Article CARD_1, MML version 4.99.1005
:: CARD_1:funcnot 1 => XBOOLE_0:func 1
notation
synonym 0 for {};
end;
:: CARD_1:attrnot 1 => CARD_1:attr 1
definition
let a1 be set;
attr a1 is cardinal means
ex b1 being ordinal set st
a1 = b1 &
(for b2 being ordinal set
st b2,b1 are_equipotent
holds b1 c= b2);
end;
:: CARD_1:dfs 1
definiens
let a1 be set;
To prove
a1 is cardinal
it is sufficient to prove
thus ex b1 being ordinal set st
a1 = b1 &
(for b2 being ordinal set
st b2,b1 are_equipotent
holds b1 c= b2);
:: CARD_1:def 1
theorem
for b1 being set holds
b1 is cardinal
iff
ex b2 being ordinal set st
b1 = b2 &
(for b3 being ordinal set
st b3,b2 are_equipotent
holds b2 c= b3);
:: CARD_1:exreg 1
registration
cluster cardinal set;
end;
:: CARD_1:modenot 1
definition
mode Cardinal is cardinal set;
end;
:: CARD_1:condreg 1
registration
cluster cardinal -> ordinal (set);
end;
:: CARD_1:th 4
theorem
for b1 being set holds
ex b2 being ordinal set st
b1,b2 are_equipotent;
:: CARD_1:prednot 1 => TARSKI:pred 1
notation
let a1, a2 be cardinal set;
synonym a1 <=` a2 for a1 c= a2;
end;
:: CARD_1:prednot 2 => HIDDEN:pred 2
notation
let a1, a2 be cardinal set;
synonym a1 <` a2 for a1 in a2;
end;
:: CARD_1:th 8
theorem
for b1, b2 being cardinal set holds
b1 = b2
iff
b1,b2 are_equipotent;
:: CARD_1:th 13
theorem
for b1, b2 being cardinal set holds
b1 in b2
iff
b1 c= b2 & b1 <> b2;
:: CARD_1:th 14
theorem
for b1, b2 being cardinal set holds
b1 in b2
iff
not b2 c= b1;
:: CARD_1:funcnot 2 => CARD_1:func 1
definition
let a1 be set;
func Card A1 -> cardinal set means
a1,it are_equipotent;
projectivity;
:: for a1 being set holds
:: Card Card a1 = Card a1;
end;
:: CARD_1:def 5
theorem
for b1 being set
for b2 being cardinal set holds
b2 = Card b1
iff
b1,b2 are_equipotent;
:: CARD_1:funcreg 1
registration
cluster {} -> cardinal;
end;
:: CARD_1:funcreg 2
registration
let a1 be empty set;
cluster Card a1 -> empty cardinal;
end;
:: CARD_1:funcreg 3
registration
let a1 be non empty set;
cluster Card a1 -> non empty cardinal;
end;
:: CARD_1:th 21
theorem
for b1, b2 being set holds
b1,b2 are_equipotent
iff
Card b1 = Card b2;
:: CARD_1:th 22
theorem
for b1 being Relation-like set
st b1 is well-ordering
holds field b1,order_type_of b1 are_equipotent;
:: CARD_1:th 23
theorem
for b1 being set
for b2 being cardinal set
st b1 c= b2
holds Card b1 c= b2;
:: CARD_1:th 24
theorem
for b1 being ordinal set holds
Card b1 c= b1;
:: CARD_1:th 25
theorem
for b1 being set
for b2 being cardinal set
st b1 in b2
holds Card b1 in b2;
:: CARD_1:th 26
theorem
for b1, b2 being set holds
Card b1 c= Card b2
iff
ex b3 being Relation-like Function-like set st
b3 is one-to-one & proj1 b3 = b1 & proj2 b3 c= b2;
:: CARD_1:th 27
theorem
for b1, b2 being set
st b1 c= b2
holds Card b1 c= Card b2;
:: CARD_1:th 28
theorem
for b1, b2 being set holds
Card b1 c= Card b2
iff
ex b3 being Relation-like Function-like set st
proj1 b3 = b2 & b1 c= proj2 b3;
:: CARD_1:th 29
theorem
for b1 being set holds
not b1,bool b1 are_equipotent;
:: CARD_1:th 30
theorem
for b1 being set holds
Card b1 in Card bool b1;
:: CARD_1:funcnot 3 => CARD_1:func 2
definition
let a1 be set;
func nextcard A1 -> cardinal set means
Card a1 in it &
(for b1 being cardinal set
st Card a1 in b1
holds it c= b1);
end;
:: CARD_1:def 6
theorem
for b1 being set
for b2 being cardinal set holds
b2 = nextcard b1
iff
Card b1 in b2 &
(for b3 being cardinal set
st Card b1 in b3
holds b2 c= b3);
:: CARD_1:th 32
theorem
for b1 being cardinal set holds
b1 in nextcard b1;
:: CARD_1:th 33
theorem
for b1 being set holds
Card {} in nextcard b1;
:: CARD_1:th 34
theorem
for b1, b2 being set
st Card b1 = Card b2
holds nextcard b1 = nextcard b2;
:: CARD_1:th 35
theorem
for b1, b2 being set
st b1,b2 are_equipotent
holds nextcard b1 = nextcard b2;
:: CARD_1:th 36
theorem
for b1 being ordinal set holds
b1 in nextcard b1;
:: CARD_1:attrnot 2 => CARD_1:attr 2
definition
let a1 be cardinal set;
attr a1 is limit means
for b1 being cardinal set holds
a1 <> nextcard b1;
end;
:: CARD_1:dfs 4
definiens
let a1 be cardinal set;
To prove
a1 is limit
it is sufficient to prove
thus for b1 being cardinal set holds
a1 <> nextcard b1;
:: CARD_1:def 7
theorem
for b1 being cardinal set holds
b1 is limit
iff
for b2 being cardinal set holds
b1 <> nextcard b2;
:: CARD_1:prednot 3 => CARD_1:attr 2
notation
let a1 be cardinal set;
synonym a1 is_limit_cardinal for limit;
end;
:: CARD_1:funcnot 4 => CARD_1:func 3
definition
let a1 be ordinal set;
func alef A1 -> set means
ex b1 being Relation-like Function-like T-Sequence-like set st
it = last b1 &
proj1 b1 = succ a1 &
b1 . {} = Card omega &
(for b2 being ordinal set
st succ b2 in succ a1
holds b1 . succ b2 = nextcard union {b1 . b2}) &
(for b2 being ordinal set
st b2 in succ a1 & b2 <> {} & b2 is being_limit_ordinal
holds b1 . b2 = Card sup (b1 | b2));
end;
:: CARD_1:def 8
theorem
for b1 being ordinal set
for b2 being set holds
b2 = alef b1
iff
ex b3 being Relation-like Function-like T-Sequence-like set st
b2 = last b3 &
proj1 b3 = succ b1 &
b3 . {} = Card omega &
(for b4 being ordinal set
st succ b4 in succ b1
holds b3 . succ b4 = nextcard union {b3 . b4}) &
(for b4 being ordinal set
st b4 in succ b1 & b4 <> {} & b4 is being_limit_ordinal
holds b3 . b4 = Card sup (b3 | b4));
:: CARD_1:funcreg 4
registration
let a1 be ordinal set;
cluster alef a1 -> cardinal;
end;
:: CARD_1:th 38
theorem
alef {} = Card omega;
:: CARD_1:th 39
theorem
for b1 being ordinal set holds
alef succ b1 = nextcard alef b1;
:: CARD_1:th 40
theorem
for b1 being ordinal set
st b1 <> {} & b1 is being_limit_ordinal
for b2 being Relation-like Function-like T-Sequence-like set
st proj1 b2 = b1 &
(for b3 being ordinal set
st b3 in b1
holds b2 . b3 = alef b3)
holds alef b1 = Card sup b2;
:: CARD_1:th 41
theorem
for b1, b2 being ordinal set holds
b1 in b2
iff
alef b1 in alef b2;
:: CARD_1:th 42
theorem
for b1, b2 being ordinal set
st alef b1 = alef b2
holds b1 = b2;
:: CARD_1:th 43
theorem
for b1, b2 being ordinal set holds
b1 c= b2
iff
alef b1 c= alef b2;
:: CARD_1:th 44
theorem
for b1, b2, b3 being set
st b1 c= b2 & b2 c= b3 & b1,b3 are_equipotent
holds b1,b2 are_equipotent & b2,b3 are_equipotent;
:: CARD_1:th 45
theorem
for b1, b2 being set
st bool b1 c= b2
holds Card b1 in Card b2 & not b1,b2 are_equipotent;
:: CARD_1:th 46
theorem
for b1 being set
st b1,{} are_equipotent
holds b1 = {};
:: CARD_1:th 47
theorem
Card {} = {};
:: CARD_1:th 48
theorem
for b1, b2 being set holds
b1,{b2} are_equipotent
iff
ex b3 being set st
b1 = {b3};
:: CARD_1:th 49
theorem
for b1, b2 being set holds
Card b1 = Card {b2}
iff
ex b3 being set st
b1 = {b3};
:: CARD_1:th 50
theorem
for b1 being set holds
Card {b1} = 1;
:: CARD_1:th 58
theorem
for b1, b2, b3, b4 being set
st b1 misses b2 & b3 misses b4 & b1,b3 are_equipotent & b2,b4 are_equipotent
holds b1 \/ b2,b3 \/ b4 are_equipotent;
:: CARD_1:th 59
theorem
for b1, b2, b3 being set
st b1 in b2 & b3 in b2
holds b2 \ {b1},b2 \ {b3} are_equipotent;
:: CARD_1:th 60
theorem
for b1 being set
for b2 being Relation-like Function-like set
st b1 c= proj1 b2 & b2 is one-to-one
holds b1,b2 .: b1 are_equipotent;
:: CARD_1:th 61
theorem
for b1, b2, b3, b4 being set
st b1,b2 are_equipotent & b3 in b1 & b4 in b2
holds b1 \ {b3},b2 \ {b4} are_equipotent;
:: CARD_1:th 62
theorem
for b1, b2 being set
st succ b1,succ b2 are_equipotent
holds b1,b2 are_equipotent;
:: CARD_1:th 63
theorem
for b1 being natural set
st b1 <> {}
holds ex b2 being natural set st
b1 = succ b2;
:: CARD_1:th 64
theorem
for b1, b2 being natural set
st b1,b2 are_equipotent
holds b1 = b2;
:: CARD_1:th 65
theorem
for b1 being set
st b1 in omega
holds b1 is cardinal;
:: CARD_1:condreg 2
registration
cluster natural -> cardinal (set);
end;
:: CARD_1:th 66
theorem
for b1 being natural set holds
b1 = Card b1;
:: CARD_1:th 68
theorem
for b1, b2 being set
st b1,b2 are_equipotent & b1 is finite
holds b2 is finite;
:: CARD_1:th 69
theorem
for b1 being natural set holds
b1 is finite & Card b1 is finite;
:: CARD_1:th 71
theorem
for b1, b2 being natural set
st Card b1 = Card b2
holds b1 = b2;
:: CARD_1:th 72
theorem
for b1, b2 being natural set holds
Card b1 c= Card b2
iff
b1 c= b2;
:: CARD_1:th 73
theorem
for b1, b2 being natural set holds
Card b1 in Card b2
iff
b1 in b2;
:: CARD_1:th 74
theorem
for b1 being set
st b1 is finite
holds ex b2 being natural set st
b1,b2 are_equipotent;
:: CARD_1:th 76
theorem
for b1 being natural set holds
nextcard Card b1 = Card succ b1;
:: CARD_1:funcnot 5 => CARD_1:func 4
definition
let a1 be natural set;
redefine func succ a1 -> Element of omega;
end;
:: CARD_1:funcnot 6 => CARD_1:func 1
notation
let a1 be finite set;
synonym card a1 for Card a1;
end;
:: CARD_1:funcnot 7 => CARD_1:func 5
definition
let a1 be finite set;
redefine func card a1 -> Element of omega;
projectivity;
:: for a1 being finite set holds
:: card card a1 = card a1;
end;
:: CARD_1:th 78
theorem
card {} = {};
:: CARD_1:th 79
theorem
for b1 being set holds
card {b1} = 1;
:: CARD_1:th 81
theorem
for b1, b2 being finite set
st b1,b2 are_equipotent
holds card b1 = card b2;
:: CARD_1:th 82
theorem
for b1 being set
st b1 is finite
holds nextcard b1 is finite;
:: CARD_1:sch 1
scheme CARD_1:sch 1
for b1 being cardinal set holds
P1[b1]
provided
P1[{}]
and
for b1 being cardinal set
st P1[b1]
holds P1[nextcard b1]
and
for b1 being cardinal set
st b1 <> {} &
b1 is limit &
(for b2 being cardinal set
st b2 in b1
holds P1[b2])
holds P1[b1];
:: CARD_1:sch 2
scheme CARD_1:sch 2
for b1 being cardinal set holds
P1[b1]
provided
for b1 being cardinal set
st for b2 being cardinal set
st b2 in b1
holds P1[b2]
holds P1[b1];
:: CARD_1:th 83
theorem
alef {} = omega;
:: CARD_1:th 84
theorem
Card omega = omega;
:: CARD_1:th 85
theorem
Card omega is limit;
:: CARD_1:condreg 3
registration
cluster -> finite (Element of omega);
end;
:: CARD_1:exreg 2
registration
cluster epsilon-transitive epsilon-connected ordinal finite cardinal set;
end;
:: CARD_1:th 86
theorem
for b1 being finite cardinal set holds
ex b2 being natural set st
b1 = Card b2;
:: CARD_1:funcreg 5
registration
let a1 be finite set;
cluster Card a1 -> finite cardinal;
end;
:: CARD_1:th 87
theorem
1 = {{}};
:: CARD_1:th 88
theorem
2 = {{},1};
:: CARD_1:th 89
theorem
3 = {{},1,2};
:: CARD_1:th 90
theorem
4 = {{},1,2,3};
:: CARD_1:th 91
theorem
5 = {{},1,2,3,4};
:: CARD_1:th 92
theorem
6 = {{},1,2,3,4,5};
:: CARD_1:th 93
theorem
7 = {{},1,2,3,4,5,6};
:: CARD_1:th 94
theorem
8 = {{},1,2,3,4,5,6,7};
:: CARD_1:th 95
theorem
9 = {{},1,2,3,4,5,6,7,8};
:: CARD_1:th 96
theorem
10 = {{},1,2,3,4,5,6,7,8,9};
:: CARD_1:th 97
theorem
for b1 being Relation-like Function-like set
st proj1 b1 is infinite & b1 is one-to-one
holds proj2 b1 is infinite;
:: CARD_1:funcnot 8 => CARD_1:func 6
definition
let a1 be natural set;
func Segm A1 -> set equals
a1;
end;
:: CARD_1:def 12
theorem
for b1 being natural set holds
Segm b1 = b1;
:: CARD_1:funcnot 9 => CARD_1:func 7
definition
let a1 be natural set;
redefine func Segm a1 -> Element of bool omega;
end;
:: CARD_1:th 102
theorem
for b1 being ordinal set
for b2 being natural set
st b1,b2 are_equipotent
holds b1 = b2;
:: CARD_1:th 103
theorem
for b1 being ordinal set holds
b1 is finite
iff
b1 in omega;
:: CARD_1:condreg 4
registration
cluster natural -> finite (set);
end;
:: CARD_1:condreg 5
registration
cluster natural -> cardinal (set);
end;
:: CARD_1:funcreg 6
registration
cluster omega -> infinite;
end;
:: CARD_1:exreg 3
registration
cluster infinite set;
end;
:: CARD_1:funcreg 7
registration
let a1 be infinite set;
cluster bool a1 -> infinite;
end;
:: CARD_1:funcreg 8
registration
let a1 be infinite set;
let a2 be non empty set;
cluster [:a1,a2:] -> infinite;
end;
:: CARD_1:funcreg 9
registration
let a1 be infinite set;
let a2 be non empty set;
cluster [:a2,a1:] -> infinite;
end;
:: CARD_1:exreg 4
registration
let a1 be infinite set;
cluster infinite Element of bool a1;
end;