:: 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
Lemma48:
for A, B being Ordinal
for x1, x2 being set st x1 <> x2 holds
( A +^ B,[:A,{x1}:] \/ [:B,{x2}:] are_equipotent & Card (A +^ B) = Card ([:A,{x1}:] \/ [:B,{x2}:]) )
deffunc H2( set , set ) -> set = [:a1,{0}:] \/ [:a2,{1}:];
Lemma58:
for X, Y being set holds
( [:X,Y:],[:Y,X:] are_equipotent & Card [:X,Y:] = Card [:Y,X:] )
definition
let M be
Cardinal;
let N be
Cardinal;
func c1 +` c2 -> Cardinal equals :: CARD_2:def 1
Card (M +^ N);
coherence
Card (M +^ N) is Cardinal
;
commutativity
for b1, M, N being Cardinal st b1 = Card (M +^ N) holds
b1 = Card (N +^ M)
func c1 *` c2 -> Cardinal equals :: CARD_2:def 2
Card [:M,N:];
coherence
Card [:M,N:] is Cardinal
;
commutativity
for b1, M, N being Cardinal st b1 = Card [:M,N:] holds
b1 = Card [:N,M:]
by ;
func exp c1,
c2 -> Cardinal equals :: CARD_2:def 3
Card (Funcs N,M);
coherence
Card (Funcs N,M) 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
X,
Y,
Z being
set holds
(
[:[:X,Y:],Z:],
[:X,[:Y,Z:]:] are_equipotent &
Card [:[:X,Y:],Z:] = Card [:X,[:Y,Z:]:] )
theorem Th13: :: CARD_2:13
Lemma61:
for X, Y being set holds [:X,Y:],[:(Card X),Y:] are_equipotent
theorem Th14: :: CARD_2:14
for
X,
Y being
set holds
(
[:X,Y:],
[:(Card X),Y:] are_equipotent &
[:X,Y:],
[:X,(Card Y):] are_equipotent &
[:X,Y:],
[:(Card X),(Card Y):] are_equipotent &
Card [:X,Y:] = Card [:(Card X),Y:] &
Card [:X,Y:] = Card [:X,(Card Y):] &
Card [:X,Y:] = Card [:(Card X),(Card Y):] )
theorem Th15: :: CARD_2:15
for
X1,
Y1,
X2,
Y2 being
set st
X1,
Y1 are_equipotent &
X2,
Y2 are_equipotent holds
(
[:X1,X2:],
[:Y1,Y2:] are_equipotent &
Card [:X1,X2:] = Card [:Y1,Y2:] )
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
canceled;
theorem Th21: :: CARD_2:21
canceled;
theorem Th22: :: CARD_2:22
theorem Th23: :: CARD_2:23
for
X1,
Y1,
X2,
Y2,
x1,
x2,
y1,
y2 being
set st
X1,
Y1 are_equipotent &
X2,
Y2 are_equipotent &
x1 <> x2 &
y1 <> y2 holds
(
[:X1,{x1}:] \/ [:X2,{x2}:],
[:Y1,{y1}:] \/ [:Y2,{y2}:] are_equipotent &
Card ([:X1,{x1}:] \/ [:X2,{x2}:]) = Card ([:Y1,{y1}:] \/ [:Y2,{y2}:]) )
theorem Th24: :: CARD_2:24
theorem Th25: :: CARD_2:25
theorem Th26: :: CARD_2:26
for
X,
Y being
set holds
(
[:X,{0}:] \/ [:Y,{1}:],
[:Y,{0}:] \/ [:X,{1}:] are_equipotent &
Card ([:X,{0}:] \/ [:Y,{1}:]) = Card ([:Y,{0}:] \/ [:X,{1}:]) )
by ;
theorem Th27: :: CARD_2:27
for
X1,
X2,
Y1,
Y2 being
set holds
(
[:X1,X2:] \/ [:Y1,Y2:],
[:X2,X1:] \/ [:Y2,Y1:] are_equipotent &
Card ([:X1,X2:] \/ [:Y1,Y2:]) = Card ([:X2,X1:] \/ [:Y2,Y1:]) )
theorem Th28: :: CARD_2:28
theorem Th29: :: CARD_2:29
Lemma78:
for x, y, X, Y being set st x <> y holds
[:X,{x}:] misses [:Y,{y}:]
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
theorem Th50: :: CARD_2:50
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
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
x1,
x2,
x3,
x4 being
set holds
card {x1,x2,x3,x4} <= 4
theorem Th72: :: CARD_2:72
for
x1,
x2,
x3,
x4,
x5 being
set holds
card {x1,x2,x3,x4,x5} <= 5
theorem Th73: :: CARD_2:73
for
x1,
x2,
x3,
x4,
x5,
x6 being
set holds
card {x1,x2,x3,x4,x5,x6} <= 6
theorem Th74: :: CARD_2:74
for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
set holds
card {x1,x2,x3,x4,x5,x6,x7} <= 7
theorem Th75: :: CARD_2:75
for
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8 being
set holds
card {x1,x2,x3,x4,x5,x6,x7,x8} <= 8
theorem Th76: :: CARD_2:76
theorem Th77: :: CARD_2:77
for
x1,
x2,
x3 being
set st
x1 <> x2 &
x1 <> x3 &
x2 <> x3 holds
card {x1,x2,x3} = 3
theorem Th78: :: CARD_2:78
for
x1,
x2,
x3,
x4 being
set st
x1 <> x2 &
x1 <> x3 &
x1 <> x4 &
x2 <> x3 &
x2 <> x4 &
x3 <> x4 holds
card {x1,x2,x3,x4} = 4
theorem Th79: :: CARD_2:79
theorem Th80: :: CARD_2:80
E118:
now
let n be
Element of
NAT ;
assume E35:
for
Z being
finite set st
card Z = n &
Z <> {} & ( for
X,
Y being
set st
X in Z &
Y in Z & not
X c= Y holds
Y c= X ) holds
union Z in Z
;
let Z be
finite set ;
assume that E36:
card Z = n + 1
and E40:
Z <> {}
and E41:
for
X,
Y being
set st
X in Z &
Y in Z & not
X c= Y holds
Y c= X
;
consider y being
Element of
Z;
per cases
( n = 0 or n <> 0 )
;
suppose E44:
n <> 0
;
set Y =
Z \ {y};
reconsider Y =
Z \ {y} as
finite set ;
{y} c= Z
by E51, ZFMISC_1:37;
then E45:
card Y =
(n + 1) - (card {y})
by ,
.=
(n + 1) - 1
by CARD_1:79
.=
n
;
for
a,
b being
set st
a in Y &
b in Y & not
a c= b holds
b c= a
by E52;
then E46:
union Y in Y
by , , , CARD_1:47;
then E47:
union Y in Z
;
E51:
(
y c= union Y or
union Y c= y )
by E52, ;
E52:
y in Z
by E51;
Z = (Z \ {y}) \/ {y}
then union Z =
(union Y) \/ (union {y})
by ZFMISC_1:96
.=
(union Y) \/ y
by ZFMISC_1:31
;
hence
union Z in Z
by , , Lemma58, XBOOLE_1:12;
end;
end;
end;
Lemma121:
for Z being finite set st Z <> {} & ( for X, Y being set st X in Z & Y in Z & not X c= Y holds
Y c= X ) holds
union Z in Z
theorem Th81: :: CARD_2:81
:: deftheorem Def4 defines are_mutually_different CARD_2:def 4 :
theorem Th82: :: CARD_2:82
for
x1,
x2,
x3,
x4,
x5 being
set st
x1,
x2,
x3,
x4,
x5 are_mutually_different holds
card {x1,x2,x3,x4,x5} = 5