Article MEMBERED, MML version 4.99.1005
:: MEMBERED:attrnot 1 => MEMBERED:attr 1
definition
let a1 be set;
attr a1 is complex-membered means
for b1 being set
st b1 in a1
holds b1 is complex;
end;
:: MEMBERED:dfs 1
definiens
let a1 be set;
To prove
a1 is complex-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is complex;
:: MEMBERED:def 1
theorem
for b1 being set holds
b1 is complex-membered
iff
for b2 being set
st b2 in b1
holds b2 is complex;
:: MEMBERED:attrnot 2 => MEMBERED:attr 2
definition
let a1 be set;
attr a1 is ext-real-membered means
for b1 being set
st b1 in a1
holds b1 is ext-real;
end;
:: MEMBERED:dfs 2
definiens
let a1 be set;
To prove
a1 is ext-real-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is ext-real;
:: MEMBERED:def 2
theorem
for b1 being set holds
b1 is ext-real-membered
iff
for b2 being set
st b2 in b1
holds b2 is ext-real;
:: MEMBERED:attrnot 3 => MEMBERED:attr 3
definition
let a1 be set;
attr a1 is real-membered means
for b1 being set
st b1 in a1
holds b1 is real;
end;
:: MEMBERED:dfs 3
definiens
let a1 be set;
To prove
a1 is real-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is real;
:: MEMBERED:def 3
theorem
for b1 being set holds
b1 is real-membered
iff
for b2 being set
st b2 in b1
holds b2 is real;
:: MEMBERED:attrnot 4 => MEMBERED:attr 4
definition
let a1 be set;
attr a1 is rational-membered means
for b1 being set
st b1 in a1
holds b1 is rational;
end;
:: MEMBERED:dfs 4
definiens
let a1 be set;
To prove
a1 is rational-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is rational;
:: MEMBERED:def 4
theorem
for b1 being set holds
b1 is rational-membered
iff
for b2 being set
st b2 in b1
holds b2 is rational;
:: MEMBERED:attrnot 5 => MEMBERED:attr 5
definition
let a1 be set;
attr a1 is integer-membered means
for b1 being set
st b1 in a1
holds b1 is integer;
end;
:: MEMBERED:dfs 5
definiens
let a1 be set;
To prove
a1 is integer-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is integer;
:: MEMBERED:def 5
theorem
for b1 being set holds
b1 is integer-membered
iff
for b2 being set
st b2 in b1
holds b2 is integer;
:: MEMBERED:attrnot 6 => MEMBERED:attr 6
definition
let a1 be set;
attr a1 is natural-membered means
for b1 being set
st b1 in a1
holds b1 is natural;
end;
:: MEMBERED:dfs 6
definiens
let a1 be set;
To prove
a1 is natural-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is natural;
:: MEMBERED:def 6
theorem
for b1 being set holds
b1 is natural-membered
iff
for b2 being set
st b2 in b1
holds b2 is natural;
:: MEMBERED:condreg 1
registration
cluster natural-membered -> integer-membered (set);
end;
:: MEMBERED:condreg 2
registration
cluster integer-membered -> rational-membered (set);
end;
:: MEMBERED:condreg 3
registration
cluster rational-membered -> real-membered (set);
end;
:: MEMBERED:condreg 4
registration
cluster real-membered -> ext-real-membered (set);
end;
:: MEMBERED:condreg 5
registration
cluster real-membered -> complex-membered (set);
end;
:: MEMBERED:exreg 1
registration
cluster non empty natural-membered set;
end;
:: MEMBERED:condreg 6
registration
cluster -> complex-membered (Element of bool COMPLEX);
end;
:: MEMBERED:condreg 7
registration
cluster -> ext-real-membered (Element of bool ExtREAL);
end;
:: MEMBERED:condreg 8
registration
cluster -> real-membered (Element of bool REAL);
end;
:: MEMBERED:condreg 9
registration
cluster -> rational-membered (Element of bool RAT);
end;
:: MEMBERED:condreg 10
registration
cluster -> integer-membered (Element of bool INT);
end;
:: MEMBERED:condreg 11
registration
cluster -> natural-membered (Element of bool NAT);
end;
:: MEMBERED:funcreg 1
registration
cluster COMPLEX -> complex-membered;
end;
:: MEMBERED:funcreg 2
registration
cluster ExtREAL -> ext-real-membered;
end;
:: MEMBERED:funcreg 3
registration
cluster REAL -> real-membered;
end;
:: MEMBERED:funcreg 4
registration
cluster RAT -> rational-membered;
end;
:: MEMBERED:funcreg 5
registration
cluster INT -> integer-membered;
end;
:: MEMBERED:funcreg 6
registration
cluster omega -> natural-membered;
end;
:: MEMBERED:th 1
theorem
for b1 being set
st b1 is complex-membered
holds b1 c= COMPLEX;
:: MEMBERED:th 2
theorem
for b1 being set
st b1 is ext-real-membered
holds b1 c= ExtREAL;
:: MEMBERED:th 3
theorem
for b1 being set
st b1 is real-membered
holds b1 c= REAL;
:: MEMBERED:th 4
theorem
for b1 being set
st b1 is rational-membered
holds b1 c= RAT;
:: MEMBERED:th 5
theorem
for b1 being set
st b1 is integer-membered
holds b1 c= INT;
:: MEMBERED:th 6
theorem
for b1 being set
st b1 is natural-membered
holds b1 c= NAT;
:: MEMBERED:condreg 12
registration
let a1 be complex-membered set;
cluster -> complex (Element of a1);
end;
:: MEMBERED:condreg 13
registration
let a1 be ext-real-membered set;
cluster -> ext-real (Element of a1);
end;
:: MEMBERED:condreg 14
registration
let a1 be real-membered set;
cluster -> real (Element of a1);
end;
:: MEMBERED:condreg 15
registration
let a1 be rational-membered set;
cluster -> rational (Element of a1);
end;
:: MEMBERED:condreg 16
registration
let a1 be integer-membered set;
cluster -> integer (Element of a1);
end;
:: MEMBERED:condreg 17
registration
let a1 be natural-membered set;
cluster -> natural (Element of a1);
end;
:: MEMBERED:th 7
theorem
for b1 being non empty complex-membered set holds
ex b2 being complex set st
b2 in b1;
:: MEMBERED:th 8
theorem
for b1 being non empty ext-real-membered set holds
ex b2 being ext-real set st
b2 in b1;
:: MEMBERED:th 9
theorem
for b1 being non empty real-membered set holds
ex b2 being real set st
b2 in b1;
:: MEMBERED:th 10
theorem
for b1 being non empty rational-membered set holds
ex b2 being rational set st
b2 in b1;
:: MEMBERED:th 11
theorem
for b1 being non empty integer-membered set holds
ex b2 being integer set st
b2 in b1;
:: MEMBERED:th 12
theorem
for b1 being non empty natural-membered set holds
ex b2 being natural set st
b2 in b1;
:: MEMBERED:th 13
theorem
for b1 being complex-membered set
st for b2 being complex set holds
b2 in b1
holds b1 = COMPLEX;
:: MEMBERED:th 14
theorem
for b1 being ext-real-membered set
st for b2 being ext-real set holds
b2 in b1
holds b1 = ExtREAL;
:: MEMBERED:th 15
theorem
for b1 being real-membered set
st for b2 being real set holds
b2 in b1
holds b1 = REAL;
:: MEMBERED:th 16
theorem
for b1 being rational-membered set
st for b2 being rational set holds
b2 in b1
holds b1 = RAT;
:: MEMBERED:th 17
theorem
for b1 being integer-membered set
st for b2 being integer set holds
b2 in b1
holds b1 = INT;
:: MEMBERED:th 18
theorem
for b1 being natural-membered set
st for b2 being natural set holds
b2 in b1
holds b1 = NAT;
:: MEMBERED:th 19
theorem
for b1 being set
for b2 being complex-membered set
st b1 c= b2
holds b1 is complex-membered;
:: MEMBERED:th 20
theorem
for b1 being set
for b2 being ext-real-membered set
st b1 c= b2
holds b1 is ext-real-membered;
:: MEMBERED:th 21
theorem
for b1 being set
for b2 being real-membered set
st b1 c= b2
holds b1 is real-membered;
:: MEMBERED:th 22
theorem
for b1 being set
for b2 being rational-membered set
st b1 c= b2
holds b1 is rational-membered;
:: MEMBERED:th 23
theorem
for b1 being set
for b2 being integer-membered set
st b1 c= b2
holds b1 is integer-membered;
:: MEMBERED:th 24
theorem
for b1 being set
for b2 being natural-membered set
st b1 c= b2
holds b1 is natural-membered;
:: MEMBERED:funcreg 7
registration
cluster {} -> natural-membered;
end;
:: MEMBERED:condreg 18
registration
cluster empty -> natural-membered (set);
end;
:: MEMBERED:funcreg 8
registration
let a1 be complex set;
cluster {a1} -> complex-membered;
end;
:: MEMBERED:funcreg 9
registration
let a1 be ext-real set;
cluster {a1} -> ext-real-membered;
end;
:: MEMBERED:funcreg 10
registration
let a1 be real set;
cluster {a1} -> real-membered;
end;
:: MEMBERED:funcreg 11
registration
let a1 be rational set;
cluster {a1} -> rational-membered;
end;
:: MEMBERED:funcreg 12
registration
let a1 be integer set;
cluster {a1} -> integer-membered;
end;
:: MEMBERED:funcreg 13
registration
let a1 be natural set;
cluster {a1} -> natural-membered;
end;
:: MEMBERED:funcreg 14
registration
let a1, a2 be complex set;
cluster {a1,a2} -> complex-membered;
end;
:: MEMBERED:funcreg 15
registration
let a1, a2 be ext-real set;
cluster {a1,a2} -> ext-real-membered;
end;
:: MEMBERED:funcreg 16
registration
let a1, a2 be real set;
cluster {a1,a2} -> real-membered;
end;
:: MEMBERED:funcreg 17
registration
let a1, a2 be rational set;
cluster {a1,a2} -> rational-membered;
end;
:: MEMBERED:funcreg 18
registration
let a1, a2 be integer set;
cluster {a1,a2} -> integer-membered;
end;
:: MEMBERED:funcreg 19
registration
let a1, a2 be natural set;
cluster {a1,a2} -> natural-membered;
end;
:: MEMBERED:funcreg 20
registration
let a1, a2, a3 be complex set;
cluster {a1,a2,a3} -> complex-membered;
end;
:: MEMBERED:funcreg 21
registration
let a1, a2, a3 be ext-real set;
cluster {a1,a2,a3} -> ext-real-membered;
end;
:: MEMBERED:funcreg 22
registration
let a1, a2, a3 be real set;
cluster {a1,a2,a3} -> real-membered;
end;
:: MEMBERED:funcreg 23
registration
let a1, a2, a3 be rational set;
cluster {a1,a2,a3} -> rational-membered;
end;
:: MEMBERED:funcreg 24
registration
let a1, a2, a3 be integer set;
cluster {a1,a2,a3} -> integer-membered;
end;
:: MEMBERED:funcreg 25
registration
let a1, a2, a3 be natural set;
cluster {a1,a2,a3} -> natural-membered;
end;
:: MEMBERED:condreg 19
registration
let a1 be complex-membered set;
cluster -> complex-membered (Element of bool a1);
end;
:: MEMBERED:condreg 20
registration
let a1 be ext-real-membered set;
cluster -> ext-real-membered (Element of bool a1);
end;
:: MEMBERED:condreg 21
registration
let a1 be real-membered set;
cluster -> real-membered (Element of bool a1);
end;
:: MEMBERED:condreg 22
registration
let a1 be rational-membered set;
cluster -> rational-membered (Element of bool a1);
end;
:: MEMBERED:condreg 23
registration
let a1 be integer-membered set;
cluster -> integer-membered (Element of bool a1);
end;
:: MEMBERED:condreg 24
registration
let a1 be natural-membered set;
cluster -> natural-membered (Element of bool a1);
end;
:: MEMBERED:funcreg 26
registration
let a1, a2 be complex-membered set;
cluster a1 \/ a2 -> complex-membered;
end;
:: MEMBERED:funcreg 27
registration
let a1, a2 be ext-real-membered set;
cluster a1 \/ a2 -> ext-real-membered;
end;
:: MEMBERED:funcreg 28
registration
let a1, a2 be real-membered set;
cluster a1 \/ a2 -> real-membered;
end;
:: MEMBERED:funcreg 29
registration
let a1, a2 be rational-membered set;
cluster a1 \/ a2 -> rational-membered;
end;
:: MEMBERED:funcreg 30
registration
let a1, a2 be integer-membered set;
cluster a1 \/ a2 -> integer-membered;
end;
:: MEMBERED:funcreg 31
registration
let a1, a2 be natural-membered set;
cluster a1 \/ a2 -> natural-membered;
end;
:: MEMBERED:funcreg 32
registration
let a1 be complex-membered set;
let a2 be set;
cluster a1 /\ a2 -> complex-membered;
end;
:: MEMBERED:funcreg 33
registration
let a1 be complex-membered set;
let a2 be set;
cluster a2 /\ a1 -> complex-membered;
end;
:: MEMBERED:funcreg 34
registration
let a1 be ext-real-membered set;
let a2 be set;
cluster a1 /\ a2 -> ext-real-membered;
end;
:: MEMBERED:funcreg 35
registration
let a1 be ext-real-membered set;
let a2 be set;
cluster a2 /\ a1 -> ext-real-membered;
end;
:: MEMBERED:funcreg 36
registration
let a1 be real-membered set;
let a2 be set;
cluster a1 /\ a2 -> real-membered;
end;
:: MEMBERED:funcreg 37
registration
let a1 be real-membered set;
let a2 be set;
cluster a2 /\ a1 -> real-membered;
end;
:: MEMBERED:funcreg 38
registration
let a1 be rational-membered set;
let a2 be set;
cluster a1 /\ a2 -> rational-membered;
end;
:: MEMBERED:funcreg 39
registration
let a1 be rational-membered set;
let a2 be set;
cluster a2 /\ a1 -> rational-membered;
end;
:: MEMBERED:funcreg 40
registration
let a1 be integer-membered set;
let a2 be set;
cluster a1 /\ a2 -> integer-membered;
end;
:: MEMBERED:funcreg 41
registration
let a1 be integer-membered set;
let a2 be set;
cluster a2 /\ a1 -> integer-membered;
end;
:: MEMBERED:funcreg 42
registration
let a1 be natural-membered set;
let a2 be set;
cluster a1 /\ a2 -> natural-membered;
end;
:: MEMBERED:funcreg 43
registration
let a1 be natural-membered set;
let a2 be set;
cluster a2 /\ a1 -> natural-membered;
end;
:: MEMBERED:funcreg 44
registration
let a1 be complex-membered set;
let a2 be set;
cluster a1 \ a2 -> complex-membered;
end;
:: MEMBERED:funcreg 45
registration
let a1 be ext-real-membered set;
let a2 be set;
cluster a1 \ a2 -> ext-real-membered;
end;
:: MEMBERED:funcreg 46
registration
let a1 be real-membered set;
let a2 be set;
cluster a1 \ a2 -> real-membered;
end;
:: MEMBERED:funcreg 47
registration
let a1 be rational-membered set;
let a2 be set;
cluster a1 \ a2 -> rational-membered;
end;
:: MEMBERED:funcreg 48
registration
let a1 be integer-membered set;
let a2 be set;
cluster a1 \ a2 -> integer-membered;
end;
:: MEMBERED:funcreg 49
registration
let a1 be natural-membered set;
let a2 be set;
cluster a1 \ a2 -> natural-membered;
end;
:: MEMBERED:funcreg 50
registration
let a1, a2 be complex-membered set;
cluster a1 \+\ a2 -> complex-membered;
end;
:: MEMBERED:funcreg 51
registration
let a1, a2 be ext-real-membered set;
cluster a1 \+\ a2 -> ext-real-membered;
end;
:: MEMBERED:funcreg 52
registration
let a1, a2 be real-membered set;
cluster a1 \+\ a2 -> real-membered;
end;
:: MEMBERED:funcreg 53
registration
let a1, a2 be rational-membered set;
cluster a1 \+\ a2 -> rational-membered;
end;
:: MEMBERED:funcreg 54
registration
let a1, a2 be integer-membered set;
cluster a1 \+\ a2 -> integer-membered;
end;
:: MEMBERED:funcreg 55
registration
let a1, a2 be natural-membered set;
cluster a1 \+\ a2 -> natural-membered;
end;
:: MEMBERED:prednot 1 => TARSKI:pred 1
definition
let a1, a2 be set;
pred A1 c= A2 means
for b1 being complex set
st b1 in a1
holds b1 in a2;
reflexivity;
:: for a1 being set holds
:: a1 c= a1;
end;
:: MEMBERED:dfs 7
definiens
let a1, a2 be complex-membered set;
To prove
a1 c= a2
it is sufficient to prove
thus for b1 being complex set
st b1 in a1
holds b1 in a2;
:: MEMBERED:def 7
theorem
for b1, b2 being complex-membered set holds
b1 c= b2
iff
for b3 being complex set
st b3 in b1
holds b3 in b2;
:: MEMBERED:prednot 2 => TARSKI:pred 1
definition
let a1, a2 be set;
pred A1 c= A2 means
for b1 being ext-real set
st b1 in a1
holds b1 in a2;
reflexivity;
:: for a1 being set holds
:: a1 c= a1;
end;
:: MEMBERED:dfs 8
definiens
let a1, a2 be ext-real-membered set;
To prove
a1 c= a2
it is sufficient to prove
thus for b1 being ext-real set
st b1 in a1
holds b1 in a2;
:: MEMBERED:def 8
theorem
for b1, b2 being ext-real-membered set holds
b1 c= b2
iff
for b3 being ext-real set
st b3 in b1
holds b3 in b2;
:: MEMBERED:prednot 3 => TARSKI:pred 1
definition
let a1, a2 be set;
pred A1 c= A2 means
for b1 being real set
st b1 in a1
holds b1 in a2;
reflexivity;
:: for a1 being set holds
:: a1 c= a1;
end;
:: MEMBERED:dfs 9
definiens
let a1, a2 be real-membered set;
To prove
a1 c= a2
it is sufficient to prove
thus for b1 being real set
st b1 in a1
holds b1 in a2;
:: MEMBERED:def 9
theorem
for b1, b2 being real-membered set holds
b1 c= b2
iff
for b3 being real set
st b3 in b1
holds b3 in b2;
:: MEMBERED:prednot 4 => TARSKI:pred 1
definition
let a1, a2 be set;
pred A1 c= A2 means
for b1 being rational set
st b1 in a1
holds b1 in a2;
reflexivity;
:: for a1 being set holds
:: a1 c= a1;
end;
:: MEMBERED:dfs 10
definiens
let a1, a2 be rational-membered set;
To prove
a1 c= a2
it is sufficient to prove
thus for b1 being rational set
st b1 in a1
holds b1 in a2;
:: MEMBERED:def 10
theorem
for b1, b2 being rational-membered set holds
b1 c= b2
iff
for b3 being rational set
st b3 in b1
holds b3 in b2;
:: MEMBERED:prednot 5 => TARSKI:pred 1
definition
let a1, a2 be set;
pred A1 c= A2 means
for b1 being integer set
st b1 in a1
holds b1 in a2;
reflexivity;
:: for a1 being set holds
:: a1 c= a1;
end;
:: MEMBERED:dfs 11
definiens
let a1, a2 be integer-membered set;
To prove
a1 c= a2
it is sufficient to prove
thus for b1 being integer set
st b1 in a1
holds b1 in a2;
:: MEMBERED:def 11
theorem
for b1, b2 being integer-membered set holds
b1 c= b2
iff
for b3 being integer set
st b3 in b1
holds b3 in b2;
:: MEMBERED:prednot 6 => TARSKI:pred 1
definition
let a1, a2 be set;
pred A1 c= A2 means
for b1 being natural set
st b1 in a1
holds b1 in a2;
reflexivity;
:: for a1 being set holds
:: a1 c= a1;
end;
:: MEMBERED:dfs 12
definiens
let a1, a2 be natural-membered set;
To prove
a1 c= a2
it is sufficient to prove
thus for b1 being natural set
st b1 in a1
holds b1 in a2;
:: MEMBERED:def 12
theorem
for b1, b2 being natural-membered set holds
b1 c= b2
iff
for b3 being natural set
st b3 in b1
holds b3 in b2;
:: MEMBERED:prednot 7 => HIDDEN:pred 1
definition
let a1, a2 be set;
pred A1 = A2 means
for b1 being complex set holds
b1 in a1
iff
b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 = a2
:: holds a2 = a1;
reflexivity;
:: for a1 being set holds
:: a1 = a1;
end;
:: MEMBERED:dfs 13
definiens
let a1, a2 be complex-membered set;
To prove
a1 = a2
it is sufficient to prove
thus for b1 being complex set holds
b1 in a1
iff
b1 in a2;
:: MEMBERED:def 13
theorem
for b1, b2 being complex-membered set holds
b1 = b2
iff
for b3 being complex set holds
b3 in b1
iff
b3 in b2;
:: MEMBERED:prednot 8 => HIDDEN:pred 1
definition
let a1, a2 be set;
pred A1 = A2 means
for b1 being ext-real set holds
b1 in a1
iff
b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 = a2
:: holds a2 = a1;
reflexivity;
:: for a1 being set holds
:: a1 = a1;
end;
:: MEMBERED:dfs 14
definiens
let a1, a2 be ext-real-membered set;
To prove
a1 = a2
it is sufficient to prove
thus for b1 being ext-real set holds
b1 in a1
iff
b1 in a2;
:: MEMBERED:def 14
theorem
for b1, b2 being ext-real-membered set holds
b1 = b2
iff
for b3 being ext-real set holds
b3 in b1
iff
b3 in b2;
:: MEMBERED:prednot 9 => HIDDEN:pred 1
definition
let a1, a2 be set;
pred A1 = A2 means
for b1 being real set holds
b1 in a1
iff
b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 = a2
:: holds a2 = a1;
reflexivity;
:: for a1 being set holds
:: a1 = a1;
end;
:: MEMBERED:dfs 15
definiens
let a1, a2 be real-membered set;
To prove
a1 = a2
it is sufficient to prove
thus for b1 being real set holds
b1 in a1
iff
b1 in a2;
:: MEMBERED:def 15
theorem
for b1, b2 being real-membered set holds
b1 = b2
iff
for b3 being real set holds
b3 in b1
iff
b3 in b2;
:: MEMBERED:prednot 10 => HIDDEN:pred 1
definition
let a1, a2 be set;
pred A1 = A2 means
for b1 being rational set holds
b1 in a1
iff
b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 = a2
:: holds a2 = a1;
reflexivity;
:: for a1 being set holds
:: a1 = a1;
end;
:: MEMBERED:dfs 16
definiens
let a1, a2 be rational-membered set;
To prove
a1 = a2
it is sufficient to prove
thus for b1 being rational set holds
b1 in a1
iff
b1 in a2;
:: MEMBERED:def 16
theorem
for b1, b2 being rational-membered set holds
b1 = b2
iff
for b3 being rational set holds
b3 in b1
iff
b3 in b2;
:: MEMBERED:prednot 11 => HIDDEN:pred 1
definition
let a1, a2 be set;
pred A1 = A2 means
for b1 being integer set holds
b1 in a1
iff
b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 = a2
:: holds a2 = a1;
reflexivity;
:: for a1 being set holds
:: a1 = a1;
end;
:: MEMBERED:dfs 17
definiens
let a1, a2 be integer-membered set;
To prove
a1 = a2
it is sufficient to prove
thus for b1 being integer set holds
b1 in a1
iff
b1 in a2;
:: MEMBERED:def 17
theorem
for b1, b2 being integer-membered set holds
b1 = b2
iff
for b3 being integer set holds
b3 in b1
iff
b3 in b2;
:: MEMBERED:prednot 12 => HIDDEN:pred 1
definition
let a1, a2 be set;
pred A1 = A2 means
for b1 being natural set holds
b1 in a1
iff
b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 = a2
:: holds a2 = a1;
reflexivity;
:: for a1 being set holds
:: a1 = a1;
end;
:: MEMBERED:dfs 18
definiens
let a1, a2 be natural-membered set;
To prove
a1 = a2
it is sufficient to prove
thus for b1 being natural set holds
b1 in a1
iff
b1 in a2;
:: MEMBERED:def 18
theorem
for b1, b2 being natural-membered set holds
b1 = b2
iff
for b3 being natural set holds
b3 in b1
iff
b3 in b2;
:: MEMBERED:prednot 13 => not XBOOLE_0:pred 1
definition
let a1, a2 be set;
pred A1 meets A2 means
for b1 being complex set
st b1 in a1
holds not b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 misses a2
:: holds a2 misses a1;
end;
:: MEMBERED:dfs 19
definiens
let a1, a2 be complex-membered set;
To prove
a1 misses a2
it is sufficient to prove
thus for b1 being complex set
st b1 in a1
holds not b1 in a2;
:: MEMBERED:def 19
theorem
for b1, b2 being complex-membered set holds
b1 misses b2
iff
for b3 being complex set
st b3 in b1
holds not b3 in b2;
:: MEMBERED:prednot 14 => not XBOOLE_0:pred 1
definition
let a1, a2 be set;
pred A1 meets A2 means
for b1 being ext-real set
st b1 in a1
holds not b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 misses a2
:: holds a2 misses a1;
end;
:: MEMBERED:dfs 20
definiens
let a1, a2 be ext-real-membered set;
To prove
a1 misses a2
it is sufficient to prove
thus for b1 being ext-real set
st b1 in a1
holds not b1 in a2;
:: MEMBERED:def 20
theorem
for b1, b2 being ext-real-membered set holds
b1 misses b2
iff
for b3 being ext-real set
st b3 in b1
holds not b3 in b2;
:: MEMBERED:prednot 15 => not XBOOLE_0:pred 1
definition
let a1, a2 be set;
pred A1 meets A2 means
for b1 being real set
st b1 in a1
holds not b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 misses a2
:: holds a2 misses a1;
end;
:: MEMBERED:dfs 21
definiens
let a1, a2 be real-membered set;
To prove
a1 misses a2
it is sufficient to prove
thus for b1 being real set
st b1 in a1
holds not b1 in a2;
:: MEMBERED:def 21
theorem
for b1, b2 being real-membered set holds
b1 misses b2
iff
for b3 being real set
st b3 in b1
holds not b3 in b2;
:: MEMBERED:prednot 16 => not XBOOLE_0:pred 1
definition
let a1, a2 be set;
pred A1 meets A2 means
for b1 being rational set
st b1 in a1
holds not b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 misses a2
:: holds a2 misses a1;
end;
:: MEMBERED:dfs 22
definiens
let a1, a2 be rational-membered set;
To prove
a1 misses a2
it is sufficient to prove
thus for b1 being rational set
st b1 in a1
holds not b1 in a2;
:: MEMBERED:def 22
theorem
for b1, b2 being rational-membered set holds
b1 misses b2
iff
for b3 being rational set
st b3 in b1
holds not b3 in b2;
:: MEMBERED:prednot 17 => not XBOOLE_0:pred 1
definition
let a1, a2 be set;
pred A1 meets A2 means
for b1 being integer set
st b1 in a1
holds not b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 misses a2
:: holds a2 misses a1;
end;
:: MEMBERED:dfs 23
definiens
let a1, a2 be integer-membered set;
To prove
a1 misses a2
it is sufficient to prove
thus for b1 being integer set
st b1 in a1
holds not b1 in a2;
:: MEMBERED:def 23
theorem
for b1, b2 being integer-membered set holds
b1 misses b2
iff
for b3 being integer set
st b3 in b1
holds not b3 in b2;
:: MEMBERED:prednot 18 => not XBOOLE_0:pred 1
definition
let a1, a2 be set;
pred A1 meets A2 means
for b1 being natural set
st b1 in a1
holds not b1 in a2;
symmetry;
:: for a1, a2 being set
:: st a1 misses a2
:: holds a2 misses a1;
end;
:: MEMBERED:dfs 24
definiens
let a1, a2 be natural-membered set;
To prove
a1 misses a2
it is sufficient to prove
thus for b1 being natural set
st b1 in a1
holds not b1 in a2;
:: MEMBERED:def 24
theorem
for b1, b2 being natural-membered set holds
b1 misses b2
iff
for b3 being natural set
st b3 in b1
holds not b3 in b2;
:: MEMBERED:th 25
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is complex-membered
holds union b1 is complex-membered;
:: MEMBERED:th 26
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is ext-real-membered
holds union b1 is ext-real-membered;
:: MEMBERED:th 27
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is real-membered
holds union b1 is real-membered;
:: MEMBERED:th 28
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is rational-membered
holds union b1 is rational-membered;
:: MEMBERED:th 29
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is integer-membered
holds union b1 is integer-membered;
:: MEMBERED:th 30
theorem
for b1 being set
st for b2 being set
st b2 in b1
holds b2 is natural-membered
holds union b1 is natural-membered;
:: MEMBERED:th 31
theorem
for b1, b2 being set
st b2 in b1 & b2 is complex-membered
holds meet b1 is complex-membered;
:: MEMBERED:th 32
theorem
for b1, b2 being set
st b2 in b1 & b2 is ext-real-membered
holds meet b1 is ext-real-membered;
:: MEMBERED:th 33
theorem
for b1, b2 being set
st b2 in b1 & b2 is real-membered
holds meet b1 is real-membered;
:: MEMBERED:th 34
theorem
for b1, b2 being set
st b2 in b1 & b2 is rational-membered
holds meet b1 is rational-membered;
:: MEMBERED:th 35
theorem
for b1, b2 being set
st b2 in b1 & b2 is integer-membered
holds meet b1 is integer-membered;
:: MEMBERED:th 36
theorem
for b1, b2 being set
st b2 in b1 & b2 is natural-membered
holds meet b1 is natural-membered;
:: MEMBERED:sch 1
scheme MEMBERED:sch 1
ex b1 being complex-membered set st
for b2 being complex set holds
b2 in b1
iff
P1[b2]
:: MEMBERED:sch 2
scheme MEMBERED:sch 2
ex b1 being ext-real-membered set st
for b2 being ext-real set holds
b2 in b1
iff
P1[b2]
:: MEMBERED:sch 3
scheme MEMBERED:sch 3
ex b1 being real-membered set st
for b2 being real set holds
b2 in b1
iff
P1[b2]
:: MEMBERED:sch 4
scheme MEMBERED:sch 4
ex b1 being rational-membered set st
for b2 being rational set holds
b2 in b1
iff
P1[b2]
:: MEMBERED:sch 5
scheme MEMBERED:sch 5
ex b1 being integer-membered set st
for b2 being integer set holds
b2 in b1
iff
P1[b2]
:: MEMBERED:sch 6
scheme MEMBERED:sch 6
ex b1 being natural-membered set st
for b2 being natural set holds
b2 in b1
iff
P1[b2]
:: MEMBERED:exreg 2
registration
cluster non empty natural-membered set;
end;
:: MEMBERED:condreg 25
registration
cluster -> natural-membered (Element of NAT);
end;