Article CLASSES1, MML version 4.99.1005
:: CLASSES1:attrnot 1 => CLASSES1:attr 1
definition
let a1 be set;
attr a1 is subset-closed means
for b1, b2 being set
st b1 in a1 & b2 c= b1
holds b2 in a1;
end;
:: CLASSES1:dfs 1
definiens
let a1 be set;
To prove
a1 is subset-closed
it is sufficient to prove
thus for b1, b2 being set
st b1 in a1 & b2 c= b1
holds b2 in a1;
:: CLASSES1:def 1
theorem
for b1 being set holds
b1 is subset-closed
iff
for b2, b3 being set
st b2 in b1 & b3 c= b2
holds b3 in b1;
:: CLASSES1:attrnot 2 => CLASSES1:attr 2
definition
let a1 be set;
attr a1 is being_Tarski-Class means
a1 is subset-closed &
(for b1 being set
st b1 in a1
holds bool b1 in a1) &
(for b1 being set
st b1 c= a1 & not b1,a1 are_equipotent
holds b1 in a1);
end;
:: CLASSES1:dfs 2
definiens
let a1 be set;
To prove
a1 is being_Tarski-Class
it is sufficient to prove
thus a1 is subset-closed &
(for b1 being set
st b1 in a1
holds bool b1 in a1) &
(for b1 being set
st b1 c= a1 & not b1,a1 are_equipotent
holds b1 in a1);
:: CLASSES1:def 2
theorem
for b1 being set holds
b1 is being_Tarski-Class
iff
b1 is subset-closed &
(for b2 being set
st b2 in b1
holds bool b2 in b1) &
(for b2 being set
st b2 c= b1 & not b2,b1 are_equipotent
holds b2 in b1);
:: CLASSES1:prednot 1 => CLASSES1:attr 2
notation
let a1 be set;
synonym a1 is_Tarski-Class for being_Tarski-Class;
end;
:: CLASSES1:prednot 2 => CLASSES1:pred 1
definition
let a1, a2 be set;
pred A2 is_Tarski-Class_of A1 means
a1 in a2 & a2 is being_Tarski-Class;
end;
:: CLASSES1:dfs 3
definiens
let a1, a2 be set;
To prove
a2 is_Tarski-Class_of a1
it is sufficient to prove
thus a1 in a2 & a2 is being_Tarski-Class;
:: CLASSES1:def 3
theorem
for b1, b2 being set holds
b2 is_Tarski-Class_of b1
iff
b1 in b2 & b2 is being_Tarski-Class;
:: CLASSES1:funcnot 1 => CLASSES1:func 1
definition
let a1 be set;
func Tarski-Class A1 -> set means
it is_Tarski-Class_of a1 &
(for b1 being set
st b1 is_Tarski-Class_of a1
holds it c= b1);
end;
:: CLASSES1:def 4
theorem
for b1, b2 being set holds
b2 = Tarski-Class b1
iff
b2 is_Tarski-Class_of b1 &
(for b3 being set
st b3 is_Tarski-Class_of b1
holds b2 c= b3);
:: CLASSES1:funcreg 1
registration
let a1 be set;
cluster Tarski-Class a1 -> non empty;
end;
:: CLASSES1:th 2
theorem
for b1 being set holds
b1 is being_Tarski-Class
iff
b1 is subset-closed &
(for b2 being set
st b2 in b1
holds bool b2 in b1) &
(for b2 being set
st b2 c= b1 & Card b2 in Card b1
holds b2 in b1);
:: CLASSES1:th 5
theorem
for b1 being set holds
b1 in Tarski-Class b1;
:: CLASSES1:th 6
theorem
for b1, b2, b3 being set
st b1 in Tarski-Class b2 & b3 c= b1
holds b3 in Tarski-Class b2;
:: CLASSES1:th 7
theorem
for b1, b2 being set
st b1 in Tarski-Class b2
holds bool b1 in Tarski-Class b2;
:: CLASSES1:th 8
theorem
for b1, b2 being set
st b1 c= Tarski-Class b2 & not b1,Tarski-Class b2 are_equipotent
holds b1 in Tarski-Class b2;
:: CLASSES1:th 9
theorem
for b1, b2 being set
st b1 c= Tarski-Class b2 & Card b1 in Card Tarski-Class b2
holds b1 in Tarski-Class b2;
:: CLASSES1:funcnot 2 => CLASSES1:func 2
definition
let a1 be set;
let a2 be ordinal set;
func Tarski-Class(A1,A2) -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a2 &
b1 . {} = {a1} &
(for b2 being ordinal set
st succ b2 in succ a2
holds b1 . succ b2 = ({b3 where b3 is Element of Tarski-Class a1: ex b4 being Element of Tarski-Class a1 st
b4 in b1 . b2 & b3 c= b4} \/ {bool b3 where b3 is Element of Tarski-Class a1: b3 in b1 . b2}) \/ ((bool (b1 . b2)) /\ Tarski-Class a1)) &
(for b2 being ordinal set
st b2 in succ a2 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = (union proj2 (b1 | b2)) /\ Tarski-Class a1);
end;
:: CLASSES1:def 5
theorem
for b1 being set
for b2 being ordinal set
for b3 being set holds
b3 = Tarski-Class(b1,b2)
iff
ex b4 being Relation-like Function-like T-Sequence-like set st
b3 = last b4 &
proj1 b4 = succ b2 &
b4 . {} = {b1} &
(for b5 being ordinal set
st succ b5 in succ b2
holds b4 . succ b5 = ({b6 where b6 is Element of Tarski-Class b1: ex b7 being Element of Tarski-Class b1 st
b7 in b4 . b5 & b6 c= b7} \/ {bool b6 where b6 is Element of Tarski-Class b1: b6 in b4 . b5}) \/ ((bool (b4 . b5)) /\ Tarski-Class b1)) &
(for b5 being ordinal set
st b5 in succ b2 & b5 <> {} & b5 is being_limit_ordinal
holds b4 . b5 = (union proj2 (b4 | b5)) /\ Tarski-Class b1);
:: CLASSES1:funcnot 3 => CLASSES1:func 3
definition
let a1 be set;
let a2 be ordinal set;
redefine func Tarski-Class(a1,a2) -> Element of bool Tarski-Class a1;
end;
:: CLASSES1:th 10
theorem
for b1 being set holds
Tarski-Class(b1,{}) = {b1};
:: CLASSES1:th 11
theorem
for b1 being set
for b2 being ordinal set holds
Tarski-Class(b1,succ b2) = ({b3 where b3 is Element of Tarski-Class b1: ex b4 being Element of Tarski-Class b1 st
b4 in Tarski-Class(b1,b2) & b3 c= b4} \/ {bool b3 where b3 is Element of Tarski-Class b1: b3 in Tarski-Class(b1,b2)}) \/ ((bool Tarski-Class(b1,b2)) /\ Tarski-Class b1);
:: CLASSES1:th 12
theorem
for b1 being set
for b2 being ordinal set
st b2 <> {} & b2 is being_limit_ordinal
holds Tarski-Class(b1,b2) = {b3 where b3 is Element of Tarski-Class b1: ex b4 being ordinal set st
b4 in b2 & b3 in Tarski-Class(b1,b4)};
:: CLASSES1:th 13
theorem
for b1, b2 being set
for b3 being ordinal set holds
b1 in Tarski-Class(b2,succ b3)
iff
((b1 c= Tarski-Class(b2,b3) implies not b1 in Tarski-Class b2) implies ex b4 being set st
b4 in Tarski-Class(b2,b3) & (b1 c= b4 or b1 = bool b4));
:: CLASSES1:th 14
theorem
for b1, b2, b3 being set
for b4 being ordinal set
st b1 c= b2 & b2 in Tarski-Class(b3,b4)
holds b1 in Tarski-Class(b3,succ b4);
:: CLASSES1:th 15
theorem
for b1, b2 being set
for b3 being ordinal set
st b1 in Tarski-Class(b2,b3)
holds bool b1 in Tarski-Class(b2,succ b3);
:: CLASSES1:th 16
theorem
for b1, b2 being set
for b3 being ordinal set
st b3 <> {} & b3 is being_limit_ordinal
holds b2 in Tarski-Class(b1,b3)
iff
ex b4 being ordinal set st
b4 in b3 & b2 in Tarski-Class(b1,b4);
:: CLASSES1:th 17
theorem
for b1, b2, b3 being set
for b4 being ordinal set
st b4 <> {} & b4 is being_limit_ordinal & b1 in Tarski-Class(b2,b4) & (b3 c= b1 or b3 = bool b1)
holds b3 in Tarski-Class(b2,b4);
:: CLASSES1:th 18
theorem
for b1 being set
for b2 being ordinal set holds
Tarski-Class(b1,b2) c= Tarski-Class(b1,succ b2);
:: CLASSES1:th 19
theorem
for b1 being set
for b2, b3 being ordinal set
st b2 c= b3
holds Tarski-Class(b1,b2) c= Tarski-Class(b1,b3);
:: CLASSES1:th 20
theorem
for b1 being set holds
ex b2 being ordinal set st
Tarski-Class(b1,b2) = Tarski-Class(b1,succ b2);
:: CLASSES1:th 21
theorem
for b1 being set
for b2 being ordinal set
st Tarski-Class(b1,b2) = Tarski-Class(b1,succ b2)
holds Tarski-Class(b1,b2) = Tarski-Class b1;
:: CLASSES1:th 22
theorem
for b1 being set holds
ex b2 being ordinal set st
Tarski-Class(b1,b2) = Tarski-Class b1;
:: CLASSES1:th 23
theorem
for b1 being set holds
ex b2 being ordinal set st
Tarski-Class(b1,b2) = Tarski-Class b1 &
(for b3 being ordinal set
st b3 in b2
holds Tarski-Class(b1,b3) <> Tarski-Class b1);
:: CLASSES1:th 24
theorem
for b1, b2 being set
st b1 <> b2 & b1 in Tarski-Class b2
holds ex b3 being ordinal set st
not b1 in Tarski-Class(b2,b3) & b1 in Tarski-Class(b2,succ b3);
:: CLASSES1:th 25
theorem
for b1 being set
st b1 is epsilon-transitive
for b2 being ordinal set
st b2 <> {}
holds Tarski-Class(b1,b2) is epsilon-transitive;
:: CLASSES1:th 26
theorem
for b1 being set holds
Tarski-Class(b1,{}) in Tarski-Class(b1,1) & Tarski-Class(b1,{}) <> Tarski-Class(b1,1);
:: CLASSES1:th 27
theorem
for b1 being set
st b1 is epsilon-transitive
holds Tarski-Class b1 is epsilon-transitive;
:: CLASSES1:th 28
theorem
for b1, b2 being set
st b1 in Tarski-Class b2
holds Card b1 in Card Tarski-Class b2;
:: CLASSES1:th 29
theorem
for b1, b2 being set
st b1 in Tarski-Class b2
holds not b1,Tarski-Class b2 are_equipotent;
:: CLASSES1:th 30
theorem
for b1, b2, b3 being set
st b2 in Tarski-Class b1 & b3 in Tarski-Class b1
holds {b2} in Tarski-Class b1 & {b2,b3} in Tarski-Class b1;
:: CLASSES1:th 31
theorem
for b1, b2, b3 being set
st b2 in Tarski-Class b1 & b3 in Tarski-Class b1
holds [b2,b3] in Tarski-Class b1;
:: CLASSES1:th 32
theorem
for b1, b2, b3 being set
st b1 c= Tarski-Class b2 & b3 c= Tarski-Class b2
holds [:b1,b3:] c= Tarski-Class b2;
:: CLASSES1:funcnot 4 => CLASSES1:func 4
definition
let a1 be ordinal set;
func Rank A1 -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a1 &
b1 . {} = {} &
(for b2 being ordinal set
st succ b2 in succ a1
holds b1 . succ b2 = bool (b1 . b2)) &
(for b2 being ordinal set
st b2 in succ a1 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = union proj2 (b1 | b2));
end;
:: CLASSES1:def 6
theorem
for b1 being ordinal set
for b2 being set holds
b2 = Rank b1
iff
ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = {} &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = bool (b3 . b4)) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = union proj2 (b3 | b4));
:: CLASSES1:th 33
theorem
Rank {} = {};
:: CLASSES1:th 34
theorem
for b1 being ordinal set holds
Rank succ b1 = bool Rank b1;
:: CLASSES1:th 35
theorem
for b1 being ordinal set
st b1 <> {} & b1 is being_limit_ordinal
for b2 being set holds
b2 in Rank b1
iff
ex b3 being ordinal set st
b3 in b1 & b2 in Rank b3;
:: CLASSES1:th 36
theorem
for b1 being set
for b2 being ordinal set holds
b1 c= Rank b2
iff
b1 in Rank succ b2;
:: CLASSES1:th 37
theorem
for b1 being ordinal set holds
Rank b1 is epsilon-transitive;
:: CLASSES1:th 38
theorem
for b1 being set
for b2 being ordinal set
st b1 in Rank b2
holds b1 c= Rank b2;
:: CLASSES1:th 39
theorem
for b1 being ordinal set holds
Rank b1 c= Rank succ b1;
:: CLASSES1:th 40
theorem
for b1 being ordinal set holds
union Rank b1 c= Rank b1;
:: CLASSES1:th 41
theorem
for b1 being set
for b2 being ordinal set
st b1 in Rank b2
holds union b1 in Rank b2;
:: CLASSES1:th 42
theorem
for b1, b2 being ordinal set holds
b1 in b2
iff
Rank b1 in Rank b2;
:: CLASSES1:th 43
theorem
for b1, b2 being ordinal set holds
b1 c= b2
iff
Rank b1 c= Rank b2;
:: CLASSES1:th 44
theorem
for b1 being ordinal set holds
b1 c= Rank b1;
:: CLASSES1:th 45
theorem
for b1 being ordinal set
for b2 being set
st b2 in Rank b1
holds not b2,Rank b1 are_equipotent & Card b2 in Card Rank b1;
:: CLASSES1:th 46
theorem
for b1 being set
for b2 being ordinal set holds
b1 c= Rank b2
iff
bool b1 c= Rank succ b2;
:: CLASSES1:th 47
theorem
for b1, b2 being set
for b3 being ordinal set
st b1 c= b2 & b2 in Rank b3
holds b1 in Rank b3;
:: CLASSES1:th 48
theorem
for b1 being set
for b2 being ordinal set holds
b1 in Rank b2
iff
bool b1 in Rank succ b2;
:: CLASSES1:th 49
theorem
for b1 being set
for b2 being ordinal set holds
b1 in Rank b2
iff
{b1} in Rank succ b2;
:: CLASSES1:th 50
theorem
for b1, b2 being set
for b3 being ordinal set holds
b1 in Rank b3 & b2 in Rank b3
iff
{b1,b2} in Rank succ b3;
:: CLASSES1:th 51
theorem
for b1, b2 being set
for b3 being ordinal set holds
b1 in Rank b3 & b2 in Rank b3
iff
[b1,b2] in Rank succ succ b3;
:: CLASSES1:th 52
theorem
for b1 being set
for b2 being ordinal set
st b1 is epsilon-transitive &
(Rank b2) /\ Tarski-Class b1 = (Rank succ b2) /\ Tarski-Class b1
holds Tarski-Class b1 c= Rank b2;
:: CLASSES1:th 53
theorem
for b1 being set
st b1 is epsilon-transitive
holds ex b2 being ordinal set st
Tarski-Class b1 c= Rank b2;
:: CLASSES1:th 54
theorem
for b1 being set
st b1 is epsilon-transitive
holds union b1 c= b1;
:: CLASSES1:th 55
theorem
for b1, b2 being set
st b1 is epsilon-transitive & b2 is epsilon-transitive
holds b1 \/ b2 is epsilon-transitive;
:: CLASSES1:th 56
theorem
for b1, b2 being set
st b1 is epsilon-transitive & b2 is epsilon-transitive
holds b1 /\ b2 is epsilon-transitive;
:: CLASSES1:funcnot 5 => CLASSES1:func 5
definition
let a1 be set;
func the_transitive-closure_of A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Relation-like Function-like set st
ex b3 being Element of NAT st
b1 in b2 . b3 &
proj1 b2 = NAT &
b2 . 0 = a1 &
(for b4 being natural set holds
b2 . (b4 + 1) = union (b2 . b4));
end;
:: CLASSES1:def 7
theorem
for b1, b2 being set holds
b2 = the_transitive-closure_of b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being Relation-like Function-like set st
ex b5 being Element of NAT st
b3 in b4 . b5 &
proj1 b4 = NAT &
b4 . 0 = b1 &
(for b6 being natural set holds
b4 . (b6 + 1) = union (b4 . b6));
:: CLASSES1:th 58
theorem
for b1 being set holds
the_transitive-closure_of b1 is epsilon-transitive;
:: CLASSES1:th 59
theorem
for b1 being set holds
b1 c= the_transitive-closure_of b1;
:: CLASSES1:th 60
theorem
for b1, b2 being set
st b1 c= b2 & b2 is epsilon-transitive
holds the_transitive-closure_of b1 c= b2;
:: CLASSES1:th 61
theorem
for b1, b2 being set
st (for b3 being set
st b1 c= b3 & b3 is epsilon-transitive
holds b2 c= b3) &
b1 c= b2 &
b2 is epsilon-transitive
holds the_transitive-closure_of b1 = b2;
:: CLASSES1:th 62
theorem
for b1 being set
st b1 is epsilon-transitive
holds the_transitive-closure_of b1 = b1;
:: CLASSES1:th 63
theorem
the_transitive-closure_of {} = {};
:: CLASSES1:th 64
theorem
for b1 being ordinal set holds
the_transitive-closure_of b1 = b1;
:: CLASSES1:th 65
theorem
for b1, b2 being set
st b1 c= b2
holds the_transitive-closure_of b1 c= the_transitive-closure_of b2;
:: CLASSES1:th 66
theorem
for b1 being set holds
the_transitive-closure_of the_transitive-closure_of b1 = the_transitive-closure_of b1;
:: CLASSES1:th 67
theorem
for b1, b2 being set holds
the_transitive-closure_of (b1 \/ b2) = (the_transitive-closure_of b1) \/ the_transitive-closure_of b2;
:: CLASSES1:th 68
theorem
for b1, b2 being set holds
the_transitive-closure_of (b1 /\ b2) c= (the_transitive-closure_of b1) /\ the_transitive-closure_of b2;
:: CLASSES1:th 69
theorem
for b1 being set holds
ex b2 being ordinal set st
b1 c= Rank b2;
:: CLASSES1:funcnot 6 => CLASSES1:func 6
definition
let a1 be set;
func the_rank_of A1 -> ordinal set means
a1 c= Rank it &
(for b1 being ordinal set
st a1 c= Rank b1
holds it c= b1);
end;
:: CLASSES1:def 8
theorem
for b1 being set
for b2 being ordinal set holds
b2 = the_rank_of b1
iff
b1 c= Rank b2 &
(for b3 being ordinal set
st b1 c= Rank b3
holds b2 c= b3);
:: CLASSES1:th 71
theorem
for b1 being set holds
the_rank_of bool b1 = succ the_rank_of b1;
:: CLASSES1:th 72
theorem
for b1 being ordinal set holds
the_rank_of Rank b1 = b1;
:: CLASSES1:th 73
theorem
for b1 being set
for b2 being ordinal set holds
b1 c= Rank b2
iff
the_rank_of b1 c= b2;
:: CLASSES1:th 74
theorem
for b1 being set
for b2 being ordinal set holds
b1 in Rank b2
iff
the_rank_of b1 in b2;
:: CLASSES1:th 75
theorem
for b1, b2 being set
st b1 c= b2
holds the_rank_of b1 c= the_rank_of b2;
:: CLASSES1:th 76
theorem
for b1, b2 being set
st b1 in b2
holds the_rank_of b1 in the_rank_of b2;
:: CLASSES1:th 77
theorem
for b1 being set
for b2 being ordinal set holds
the_rank_of b1 c= b2
iff
for b3 being set
st b3 in b1
holds the_rank_of b3 in b2;
:: CLASSES1:th 78
theorem
for b1 being set
for b2 being ordinal set holds
b2 c= the_rank_of b1
iff
for b3 being ordinal set
st b3 in b2
holds ex b4 being set st
b4 in b1 & b3 c= the_rank_of b4;
:: CLASSES1:th 79
theorem
for b1 being set holds
the_rank_of b1 = {}
iff
b1 = {};
:: CLASSES1:th 80
theorem
for b1 being set
for b2 being ordinal set
st the_rank_of b1 = succ b2
holds ex b3 being set st
b3 in b1 & the_rank_of b3 = b2;
:: CLASSES1:th 81
theorem
for b1 being ordinal set holds
the_rank_of b1 = b1;
:: CLASSES1:th 82
theorem
for b1 being set holds
the_rank_of Tarski-Class b1 <> {} & the_rank_of Tarski-Class b1 is being_limit_ordinal;
:: CLASSES1:sch 1
scheme CLASSES1:sch 1
{F1 -> set}:
ex b1 being Relation-like Function-like set st
proj1 b1 = F1() &
(for b2 being set
st b2 in F1()
holds P1[b2, b1 . b2])
provided
for b1 being set
st b1 in F1()
holds ex b2 being set st
P1[b1, b2];