:: CARD_2 semantic presentation
theorem Th1: :: CARD_2:1
canceled;
theorem Th2: :: CARD_2:2
theorem Th3: :: CARD_2:3
theorem Th4: :: CARD_2:4
theorem Th5: :: CARD_2:5
theorem Th6: :: CARD_2:6
deffunc H1( set ) -> set = a1 `1 ;
theorem Th7: :: CARD_2:7
Lemma3:
for b1, b2 being Ordinal
for b3, b4 being set holds
( b3 <> b4 implies ( b1 +^ b2,[:b1,{b3}:] \/ [:b2,{b4}:] are_equipotent & Card (b1 +^ b2) = Card ([:b1,{b3}:] \/ [:b2,{b4}:]) ) )
deffunc H2( set , set ) -> set = [:a1,{0}:] \/ [:a2,{1}:];
Lemma4:
for b1, b2 being set holds
( [:b1,b2:],[:b2,b1:] are_equipotent & Card [:b1,b2:] = Card [:b2,b1:] )
definition
let c
1, c
2 be
Cardinal;
func c
1 +` c
2 -> Cardinal equals :: CARD_2:def 1
Card (a1 +^ a2);
coherence
Card (c1 +^ c2) is Cardinal
;
commutativity
for b1, b2, b3 being Cardinal holds
( b1 = Card (b2 +^ b3) implies b1 = Card (b3 +^ b2) )
func c
1 *` c
2 -> Cardinal equals :: CARD_2:def 2
Card [:a1,a2:];
coherence
Card [:c1,c2:] is Cardinal
;
commutativity
for b1, b2, b3 being Cardinal holds
( b1 = Card [:b2,b3:] implies b1 = Card [:b3,b2:] )
by Lemma4;
func exp c
1,c
2 -> Cardinal equals :: CARD_2:def 3
Card (Funcs a2,a1);
coherence
Card (Funcs c2,c1) is Cardinal
;
end;
:: deftheorem Def1 defines +` CARD_2:def 1 :
:: deftheorem Def2 defines *` CARD_2:def 2 :
:: deftheorem Def3 defines exp CARD_2:def 3 :
theorem Th8: :: CARD_2:8
canceled;
theorem Th9: :: CARD_2:9
canceled;
theorem Th10: :: CARD_2:10
canceled;
theorem Th11: :: CARD_2:11
theorem Th12: :: CARD_2:12
for b
1, b
2, b
3 being
set holds
(
[:[:b1,b2:],b3:],
[:b1,[:b2,b3:]:] are_equipotent &
Card [:[:b1,b2:],b3:] = Card [:b1,[:b2,b3:]:] )
theorem Th13: :: CARD_2:13
Lemma7:
for b1, b2 being set holds [:b1,b2:],[:(Card b1),b2:] are_equipotent
theorem Th14: :: CARD_2:14
for b
1, b
2 being
set holds
(
[:b1,b2:],
[:(Card b1),b2:] are_equipotent &
[:b1,b2:],
[:b1,(Card b2):] are_equipotent &
[:b1,b2:],
[:(Card b1),(Card b2):] are_equipotent &
Card [:b1,b2:] = Card [:(Card b1),b2:] &
Card [:b1,b2:] = Card [:b1,(Card b2):] &
Card [:b1,b2:] = Card [:(Card b1),(Card b2):] )
theorem Th15: :: CARD_2:15
for b
1, b
2, b
3, b
4 being
set holds
( b
1,b
2 are_equipotent & b
3,b
4 are_equipotent implies (
[:b1,b3:],
[:b2,b4:] are_equipotent &
Card [:b1,b3:] = Card [:b2,b4:] ) )
theorem Th16: :: CARD_2:16
theorem Th17: :: CARD_2:17
theorem Th18: :: CARD_2:18
deffunc H3( set , set ) -> set = [:a1,{0}:] \/ [:a2,{1}:];
deffunc H4( set , set , set , set ) -> set = [:a1,{a3}:] \/ [:a2,{a4}:];
theorem Th19: :: CARD_2:19
theorem Th20: :: CARD_2:20
theorem Th21: :: CARD_2:21
canceled;
theorem Th22: :: CARD_2:22
theorem Th23: :: CARD_2:23
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( b
1,b
2 are_equipotent & b
3,b
4 are_equipotent & b
5 <> b
6 & b
7 <> b
8 implies (
[:b1,{b5}:] \/ [:b3,{b6}:],
[:b2,{b7}:] \/ [:b4,{b8}:] are_equipotent &
Card ([:b1,{b5}:] \/ [:b3,{b6}:]) = Card ([:b2,{b7}:] \/ [:b4,{b8}:]) ) )
theorem Th24: :: CARD_2:24
theorem Th25: :: CARD_2:25
theorem Th26: :: CARD_2:26
for b
1, b
2 being
set holds
(
[:b1,{0}:] \/ [:b2,{1}:],
[:b2,{0}:] \/ [:b1,{1}:] are_equipotent &
Card ([:b1,{0}:] \/ [:b2,{1}:]) = Card ([:b2,{0}:] \/ [:b1,{1}:]) )
by Th23;
theorem Th27: :: CARD_2:27
for b
1, b
2, b
3, b
4 being
set holds
(
[:b1,b2:] \/ [:b3,b4:],
[:b2,b1:] \/ [:b4,b3:] are_equipotent &
Card ([:b1,b2:] \/ [:b3,b4:]) = Card ([:b2,b1:] \/ [:b4,b3:]) )
theorem Th28: :: CARD_2:28
theorem Th29: :: CARD_2:29
Lemma18:
for b1, b2, b3, b4 being set holds
( b1 <> b2 implies [:b3,{b1}:] misses [:b4,{b2}:] )
theorem Th30: :: CARD_2:30
canceled;
theorem Th31: :: CARD_2:31
theorem Th32: :: CARD_2:32
theorem Th33: :: CARD_2:33
theorem Th34: :: CARD_2:34
canceled;
theorem Th35: :: CARD_2:35
theorem Th36: :: CARD_2:36
theorem Th37: :: CARD_2:37
theorem Th38: :: CARD_2:38
theorem Th39: :: CARD_2:39
theorem Th40: :: CARD_2:40
theorem Th41: :: CARD_2:41
theorem Th42: :: CARD_2:42
theorem Th43: :: CARD_2:43
theorem Th44: :: CARD_2:44
theorem Th45: :: CARD_2:45
theorem Th46: :: CARD_2:46
theorem Th47: :: CARD_2:47
theorem Th48: :: CARD_2:48
theorem Th49: :: CARD_2:49
for b
1, b
2 being
Nat holds b
1 + b
2 = b
1 +^ b
2
theorem Th50: :: CARD_2:50
for b
1, b
2 being
Nat holds b
1 * b
2 = b
1 *^ b
2
theorem Th51: :: CARD_2:51
theorem Th52: :: CARD_2:52
theorem Th53: :: CARD_2:53
theorem Th54: :: CARD_2:54
theorem Th55: :: CARD_2:55
canceled;
theorem Th56: :: CARD_2:56
canceled;
theorem Th57: :: CARD_2:57
theorem Th58: :: CARD_2:58
theorem Th59: :: CARD_2:59
theorem Th60: :: CARD_2:60
for b
1 being
set holds
(
Card b
1 = 1 iff ex b
2 being
set st b
1 = {b2} )
theorem Th61: :: CARD_2:61
theorem Th62: :: CARD_2:62
theorem Th63: :: CARD_2:63
theorem Th64: :: CARD_2:64
theorem Th65: :: CARD_2:65
theorem Th66: :: CARD_2:66
theorem Th67: :: CARD_2:67
theorem Th68: :: CARD_2:68
theorem Th69: :: CARD_2:69
theorem Th70: :: CARD_2:70
theorem Th71: :: CARD_2:71
for b
1, b
2, b
3, b
4 being
set holds
card {b1,b2,b3,b4} <= 4
theorem Th72: :: CARD_2:72
for b
1, b
2, b
3, b
4, b
5 being
set holds
card {b1,b2,b3,b4,b5} <= 5
theorem Th73: :: CARD_2:73
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
card {b1,b2,b3,b4,b5,b6} <= 6
theorem Th74: :: CARD_2:74
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
card {b1,b2,b3,b4,b5,b6,b7} <= 7
theorem Th75: :: CARD_2:75
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
card {b1,b2,b3,b4,b5,b6,b7,b8} <= 8
theorem Th76: :: CARD_2:76
for b
1, b
2 being
set holds
( b
1 <> b
2 implies
card {b1,b2} = 2 )
theorem Th77: :: CARD_2:77
for b
1, b
2, b
3 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
2 <> b
3 implies
card {b1,b2,b3} = 3 )
theorem Th78: :: CARD_2:78
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 implies
card {b1,b2,b3,b4} = 4 )
theorem Th79: :: CARD_2:79
for b
1 being
finite set holds
not (
card b
1 = 2 & ( for b
2, b
3 being
set holds
not ( b
2 <> b
3 & b
1 = {b2,b3} ) ) )
theorem Th80: :: CARD_2:80
E48:
now
let c
1 be
Nat;
assume E49:
for b
1 being
finite set holds
(
card b
1 = c
1 & b
1 <> {} & ( for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & not b
2 c= b
3 & not b
3 c= b
2 ) ) implies
union b
1 in b
1 )
;
let c
2 be
finite set ;
assume that E50:
card c
2 = c
1 + 1
and E51:
c
2 <> {}
and E52:
for b
1, b
2 being
set holds
not ( b
1 in c
2 & b
2 in c
2 & not b
1 c= b
2 & not b
2 c= b
1 )
;
consider c
3 being
Element of c
2;
per cases
not ( not c1 = 0 & not c1 <> 0 )
;
suppose E53:
c
1 <> 0
;
set c
4 = c
2 \ {c3};
reconsider c
5 = c
2 \ {c3} as
finite set ;
{c3} c= c
2
by E51, ZFMISC_1:37;
then E54:
card c
5 =
(c1 + 1) - (card {c3})
by E50, Th63
.=
(c1 + 1) - 1
by CARD_1:79
.=
c
1
;
for b
1, b
2 being
set holds
not ( b
1 in c
5 & b
2 in c
5 & not b
1 c= b
2 & not b
2 c= b
1 )
by E52;
then E55:
union c
5 in c
5
by E49, E53, E54, CARD_1:47;
then E56:
union c
5 in c
2
;
E57:
( c
3 c= union c
5 or
union c
5 c= c
3 )
by E52, E55;
E58:
c
3 in c
2
by E51;
c
2 = (c2 \ {c3}) \/ {c3}
then union c
2 =
(union c5) \/ (union {c3})
by ZFMISC_1:96
.=
(union c5) \/ c
3
by ZFMISC_1:31
;
hence
union c
2 in c
2
by E56, E57, E58, XBOOLE_1:12;
end;
end;
end;
Lemma49:
for b1 being finite set holds
( b1 <> {} & ( for b2, b3 being set holds
not ( b2 in b1 & b3 in b1 & not b2 c= b3 & not b3 c= b2 ) ) implies union b1 in b1 )
theorem Th81: :: CARD_2:81
:: deftheorem Def4 defines are_mutually_different CARD_2:def 4 :
theorem Th82: :: CARD_2:82
for b
1, b
2, b
3, b
4, b
5 being
set holds
( b
1,b
2,b
3,b
4,b
5 are_mutually_different implies
card {b1,b2,b3,b4,b5} = 5 )