:: Cardinal Numbers
:: by Grzegorz Bancerek
::
:: Received September 19, 1989
:: Copyright (c) 1990-2012 Association of Mizar Users


begin

notation
synonym 0 for {} ;
end;

definition
let IT be set ;
attr IT is cardinal means :Def1: :: CARD_1:def 1
 ex B being Ordinal st
( IT = B & ( for A being Ordinal st A,B are_equipotent holds
B c= A ) );
end;

:: deftheorem Def1 defines cardinal CARD_1:def 1 :
for IT being set holds
( IT is cardinal iff ex B being Ordinal st
( IT = B & ( for A being Ordinal st A,B are_equipotent holds
B c= A ) ) );

registration
cluster cardinal for set ;
existence
ex b1 being set st b1 is cardinal
proof end;
end;

definition
mode Cardinal is cardinal set ;
end;

registration
cluster cardinal -> ordinal for set ;
coherence
for b1 being set st b1 is cardinal holds
b1 is ordinal
proof end;
end;

theorem Th1: :: CARD_1:1
for X being set ex A being Ordinal st X,A are_equipotent
proof end;

theorem Th2: :: CARD_1:2
for M, N being Cardinal holds
( M = N iff M,N are_equipotent )
proof end;

theorem :: CARD_1:3
for M, N being Cardinal holds
( M in N iff ( M c= N & M <> N ) )
proof end;

theorem :: CARD_1:4
for M, N being Cardinal holds
( M in N iff not N c= M ) by ORDINAL1:5, ORDINAL1:16;

definition
let X be set ;
func card X -> Cardinal means :Def2: :: CARD_1:def 2
X,it are_equipotent ;
existence
ex b1 being Cardinal st X,b1 are_equipotent
proof end;
uniqueness
for b1, b2 being Cardinal st X,b1 are_equipotent & X,b2 are_equipotent holds
b1 = b2
proof end;
projectivity
for b1 being Cardinal
for b2 being set st b2,b1 are_equipotent holds
b1,b1 are_equipotent ;
end;

:: deftheorem Def2 defines card CARD_1:def 2 :
for X being set
for b2 being Cardinal holds
( b2 = card X iff X,b2 are_equipotent );

registration
cluster empty -> cardinal for set ;
coherence
for b1 being set st b1 is empty holds
b1 is cardinal
proof end;
end;

registration
let X be empty set ;
cluster card X -> empty ;
coherence
card X is empty by Def2;
end;

registration
let X be non empty set ;
cluster card X -> non empty ;
coherence
not card X is empty
proof end;
end;

theorem Th5: :: CARD_1:5
for X, Y being set holds
( X,Y are_equipotent iff card X = card Y )
proof end;

theorem Th6: :: CARD_1:6
for R being Relation st R is well-ordering holds
field R, order_type_of R are_equipotent
proof end;

theorem Th7: :: CARD_1:7
for X being set
for M being Cardinal st X c= M holds
card X c= M
proof end;

theorem Th8: :: CARD_1:8
for A being Ordinal holds card A c= A
proof end;

theorem :: CARD_1:9
for X being set
for M being Cardinal st X in M holds
card X in M by Th8, ORDINAL1:12;

:: WP: Cantor-Bernstein Theorem
theorem Th10: :: CARD_1:10
for X, Y being set holds
( card X c= card Y iff ex f being Function st
( f is one-to-one & dom f = X & rng f c= Y ) )
proof end;

theorem Th11: :: CARD_1:11
for X, Y being set st X c= Y holds
card X c= card Y
proof end;

theorem Th12: :: CARD_1:12
for X, Y being set holds
( card X c= card Y iff ex f being Function st
( dom f = Y & X c= rng f ) )
proof end;

theorem Th13: :: CARD_1:13
for X being set holds not X, bool X are_equipotent
proof end;

:: WP: Cantor Theorem
theorem Th14: :: CARD_1:14
for X being set holds card X in card (bool X)
proof end;

definition
let X be set ;
func nextcard X -> Cardinal means :Def3: :: CARD_1:def 3
( card X in it & ( for M being Cardinal st card X in M holds
it c= M ) );
existence
ex b1 being Cardinal st
( card X in b1 & ( for M being Cardinal st card X in M holds
b1 c= M ) )
proof end;
uniqueness
for b1, b2 being Cardinal st card X in b1 & ( for M being Cardinal st card X in M holds
b1 c= M ) & card X in b2 & ( for M being Cardinal st card X in M holds
b2 c= M ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines nextcard CARD_1:def 3 :
for X being set
for b2 being Cardinal holds
( b2 = nextcard X iff ( card X in b2 & ( for M being Cardinal st card X in M holds
b2 c= M ) ) );

theorem :: CARD_1:15
for X being set holds {} in nextcard X
proof end;

theorem Th16: :: CARD_1:16
for X, Y being set st card X = card Y holds
nextcard X = nextcard Y
proof end;

theorem Th17: :: CARD_1:17
for X, Y being set st X,Y are_equipotent holds
nextcard X = nextcard Y
proof end;

theorem Th18: :: CARD_1:18
for A being Ordinal holds A in nextcard A
proof end;

definition
let M be Cardinal;
attr M is limit_cardinal means :Def4: :: CARD_1:def 4
 for N being Cardinal holds not M = nextcard N;
end;

:: deftheorem Def4 defines limit_cardinal CARD_1:def 4 :
for M being Cardinal holds
( M is limit_cardinal iff for N being Cardinal holds not M = nextcard N );

definition
let A be Ordinal;
func alef A -> set means :Def5: :: CARD_1:def 5
 ex S being T-Sequence st
( it = last S & dom S = succ A & S . {} = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (union {(S . B)}) ) & ( for B being Ordinal st B in succ A & B <> {} & B is limit_ordinal holds
S . B = card (sup (S | B)) ) );
correctness
existence
ex b1 being set ex S being T-Sequence st
( b1 = last S & dom S = succ A & S . {} = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (union {(S . B)}) ) & ( for B being Ordinal st B in succ A & B <> {} & B is limit_ordinal holds
S . B = card (sup (S | B)) ) );
uniqueness
for b1, b2 being set st ex S being T-Sequence st
( b1 = last S & dom S = succ A & S . {} = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (union {(S . B)}) ) & ( for B being Ordinal st B in succ A & B <> {} & B is limit_ordinal holds
S . B = card (sup (S | B)) ) ) & ex S being T-Sequence st
( b2 = last S & dom S = succ A & S . {} = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (union {(S . B)}) ) & ( for B being Ordinal st B in succ A & B <> {} & B is limit_ordinal holds
S . B = card (sup (S | B)) ) ) holds
b1 = b2;
proof end;
end;

:: deftheorem Def5 defines alef CARD_1:def 5 :
for A being Ordinal
for b2 being set holds
( b2 = alef A iff ex S being T-Sequence st
( b2 = last S & dom S = succ A & S . {} = card omega & ( for B being Ordinal st succ B in succ A holds
S . (succ B) = nextcard (union {(S . B)}) ) & ( for B being Ordinal st B in succ A & B <> {} & B is limit_ordinal holds
S . B = card (sup (S | B)) ) ) );

Lm1: now :: thesis: ( alef 0 = card omega & ( for A being Ordinal holds H3( succ A) = H2(A,H3(A)) ) & ( for A being Ordinal st A <> {} & A is limit_ordinal holds
for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S) ) )
deffunc H1( Ordinal, T-Sequence) -> Cardinal = card (sup $2);
deffunc H2( Ordinal, set ) -> Cardinal = nextcard (union {$2});
deffunc H3( Ordinal) -> set = alef $1;
A1: for A being Ordinal
for x being set holds
( x = H3(A) iff ex S being T-Sequence st
( x = last S & dom S = succ A & S . {} = card omega & ( for C being Ordinal st succ C in succ A holds
S . (succ C) = H2(C,S . C) ) & ( for C being Ordinal st C in succ A & C <> {} & C is limit_ordinal holds
S . C = H1(C,S | C) ) ) ) by Def5;
H3( {} ) = card omega from ORDINAL2:sch 8(A1);
hence alef 0 = card omega ; :: thesis: ( ( for A being Ordinal holds H3( succ A) = H2(A,H3(A)) ) & ( for A being Ordinal st A <> {} & A is limit_ordinal holds
for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S) ) )
thus for A being Ordinal holds H3( succ A) = H2(A,H3(A)) from ORDINAL2:sch 9(A1); :: thesis: for A being Ordinal st A <> {} & A is limit_ordinal holds
for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S)
thus for A being Ordinal st A <> {} & A is limit_ordinal holds
for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S) :: thesis: verum
proof
let A be Ordinal; :: thesis: ( A <> {} & A is limit_ordinal implies for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S) )
assume A2: ( A <> {} & A is limit_ordinal ) ; :: thesis: for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S)
let S be T-Sequence; :: thesis: ( dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) implies alef A = card (sup S) )
assume that
A3: dom S = A and
A4: for B being Ordinal st B in A holds
S . B = H3(B) ; :: thesis: alef A = card (sup S)
thus H3(A) = H1(A,S) from ORDINAL2:sch 10(A1, A2, A3, A4); :: thesis: verum
end;
end;

deffunc H1( Ordinal) -> set = alef $1;

registration
let A be Ordinal;
cluster alef A -> cardinal ;
coherence
alef A is cardinal
proof end;
end;

theorem Th19: :: CARD_1:19
for A being Ordinal holds alef (succ A) = nextcard (alef A)
proof end;

theorem :: CARD_1:20
for A being Ordinal st A <> {} & A is limit_ordinal holds
for S being T-Sequence st dom S = A & ( for B being Ordinal st B in A holds
S . B = alef B ) holds
alef A = card (sup S) by Lm1;

theorem Th21: :: CARD_1:21
for A, B being Ordinal holds
( A in B iff alef A in alef B )
proof end;

theorem Th22: :: CARD_1:22
for A, B being Ordinal st alef A = alef B holds
A = B
proof end;

theorem :: CARD_1:23
for A, B being Ordinal holds
( A c= B iff alef A c= alef B )
proof end;

theorem :: CARD_1:24
for X, Y, Z being set st X c= Y & Y c= Z & X,Z are_equipotent holds
( X,Y are_equipotent & Y,Z are_equipotent )
proof end;

theorem :: CARD_1:25
for Y, X being set st bool Y c= X holds
( card Y in card X & not Y,X are_equipotent )
proof end;

theorem Th26: :: CARD_1:26
for X being set st X, {} are_equipotent holds
X = {}
proof end;

theorem :: CARD_1:27
card {} = {} ;

theorem Th28: :: CARD_1:28
for X, x being set holds
( X,{x} are_equipotent iff ex x being set st X = {x} )
proof end;

theorem Th29: :: CARD_1:29
for X, x being set holds
( card X = card {x} iff ex x being set st X = {x} )
proof end;

theorem Th30: :: CARD_1:30
for x being set holds card {x} = 1
proof end;

theorem Th31: :: CARD_1:31
for X, X1, Y, Y1 being set st X misses X1 & Y misses Y1 & X,Y are_equipotent & X1,Y1 are_equipotent holds
X \/ X1,Y \/ Y1 are_equipotent
proof end;

theorem Th32: :: CARD_1:32
for x, X, y being set st x in X & y in X holds
X \ {x},X \ {y} are_equipotent
proof end;

theorem Th33: :: CARD_1:33
for X being set
for f being Function st X c= dom f & f is one-to-one holds
X,f .: X are_equipotent
proof end;

theorem Th34: :: CARD_1:34
for X, Y, x, y being set st X,Y are_equipotent & x in X & y in Y holds
X \ {x},Y \ {y} are_equipotent
proof end;

theorem Th35: :: CARD_1:35
for X, Y being set st succ X, succ Y are_equipotent holds
X,Y are_equipotent
proof end;

theorem Th36: :: CARD_1:36
for n being Nat holds
( n = {} or ex m being Nat st n = succ m )
proof end;

Lm2: for n, m being Nat st n,m are_equipotent holds
n = m
proof end;

theorem Th37: :: CARD_1:37
for x being set st x in omega holds
x is cardinal
proof end;

registration
cluster natural -> cardinal for set ;
correctness
coherence
for b1 being number st b1 is natural holds
b1 is cardinal ;
proof end;
end;

theorem Th38: :: CARD_1:38
for X, Y being set st X,Y are_equipotent & X is finite holds
Y is finite
proof end;

theorem Th39: :: CARD_1:39
for n being Nat holds
( n is finite & card n is finite )
proof end;

theorem :: CARD_1:40
for n, m being Nat st card n = card m holds
n = m
proof end;

theorem :: CARD_1:41
for n, m being Nat holds
( card n c= card m iff n c= m )
proof end;

theorem Th42: :: CARD_1:42
for n, m being Nat holds
( card n in card m iff n in m )
proof end;

theorem Th43: :: CARD_1:43
for X being set st X is finite holds
ex n being Nat st X,n are_equipotent
proof end;

theorem Th44: :: CARD_1:44
for n being Nat holds nextcard (card n) = card (succ n)
proof end;

definition
let n be Nat;
:: original: succ
redefine func succ n -> Element of omega ;
coherence
succ n is Element of omega by ORDINAL1:def 12;
end;

definition
let X be finite set ;
:: original: card
redefine func card X -> Element of omega ;
coherence
card X is Element of omega
proof end;
end;

theorem :: CARD_1:45
for X being set st X is finite holds
nextcard X is finite
proof end;

scheme :: CARD_1:sch 1
CardinalInd{ P1[ set ] } :
for M being Cardinal holds P1[M]
provided
A1: P1[ {} ] and
A2: for M being Cardinal st P1[M] holds
P1[ nextcard M] and
A3: for M being Cardinal st M <> {} & M is limit_cardinal & ( for N being Cardinal st N in M holds
P1[N] ) holds
P1[M]
proof end;

scheme :: CARD_1:sch 2
CardinalCompInd{ P1[ set ] } :
for M being Cardinal holds P1[M]
provided
A1: for M being Cardinal st ( for N being Cardinal st N in M holds
P1[N] ) holds
P1[M]
proof end;

theorem Th46: :: CARD_1:46
alef 0 = omega
proof end;

registration
cluster omega -> cardinal for number ;
coherence
for b1 being number st b1 = omega holds
b1 is cardinal by Th46;
end;

theorem :: CARD_1:47
card omega = omega by Lm1, Th46;

registration
cluster omega -> limit_cardinal ;
coherence
omega is limit_cardinal
proof end;
end;

registration
cluster -> finite for Element of omega ;
coherence
for b1 being Element of omega holds b1 is finite by Th39;
end;

registration
cluster epsilon-transitive epsilon-connected ordinal finite cardinal for set ;
existence
ex b1 being Cardinal st b1 is finite
proof end;
end;

theorem :: CARD_1:48
for M being finite Cardinal ex n being Nat st M = card n
proof end;

registration
let X be finite set ;
cluster card X -> finite ;
coherence
card X is finite ;
end;

Lm3: for A being Ordinal
for n being Nat st A,n are_equipotent holds
A = n
proof end;

Lm4: for A being Ordinal holds
( A is finite iff A in omega )
proof end;

registration
cluster omega -> infinite ;
coherence
not omega is finite
proof end;
end;

registration
cluster infinite for set ;
existence
ex b1 being set st b1 is infinite
proof end;
end;

registration
let X be infinite set ;
cluster card X -> infinite ;
coherence
not card X is finite
proof end;
end;

begin

theorem Th49: :: CARD_1:49
1 = {0}
proof end;

theorem Th50: :: CARD_1:50
2 = {0,1}
proof end;

theorem Th51: :: CARD_1:51
3 = {0,1,2}
proof end;

theorem Th52: :: CARD_1:52
4 = {0,1,2,3}
proof end;

theorem Th53: :: CARD_1:53
5 = {0,1,2,3,4}
proof end;

theorem Th54: :: CARD_1:54
6 = {0,1,2,3,4,5}
proof end;

theorem Th55: :: CARD_1:55
7 = {0,1,2,3,4,5,6}
proof end;

theorem Th56: :: CARD_1:56
8 = {0,1,2,3,4,5,6,7}
proof end;

theorem Th57: :: CARD_1:57
9 = {0,1,2,3,4,5,6,7,8}
proof end;

theorem :: CARD_1:58
10 = {0,1,2,3,4,5,6,7,8,9}
proof end;

:: Moved from FRECHET2 by AK on 27.12.2006
theorem :: CARD_1:59
for f being Function st dom f is infinite & f is one-to-one holds
rng f is infinite
proof end;

definition
let n be Nat;
func Segm n -> set equals :: CARD_1:def 6
n;
coherence
n is set ;
end;

:: deftheorem defines Segm CARD_1:def 6 :
for n being Nat holds Segm n = n;

definition
let n be Nat;
:: original: Segm
redefine func Segm n -> Subset of omega;
coherence
Segm n is Subset of omega
proof end;
end;

:: from CARD_4, 2007.08.16, A.T.
theorem :: CARD_1:60
for A being Ordinal
for n being Nat st A,n are_equipotent holds
A = n by Lm3;

theorem :: CARD_1:61
for A being Ordinal holds
( A is finite iff A in omega ) by Lm4;

registration
cluster natural -> finite for set ;
coherence
for b1 being set st b1 is natural holds
b1 is finite
proof end;
end;

:: from CARD_4, CARD_5 etc. 2008.02.21, A.T.
registration
let A be infinite set ;
cluster bool A -> infinite ;
coherence
not bool A is finite
proof end;
let B be non empty set ;
cluster [:A,B:] -> infinite ;
coherence
not [:A,B:] is finite
proof end;
cluster [:B,A:] -> infinite ;
coherence
not [:B,A:] is finite
proof end;
end;

registration
let X be infinite set ;
cluster infinite for Element of bool X;
existence
ex b1 being Subset of X st b1 is infinite
proof end;
end;

:: from NECKLA_2, 2008.06.28, A.T.
registration
cluster ordinal finite -> natural for set ;
coherence
for b1 being number st b1 is finite & b1 is ordinal holds
b1 is natural
proof end;
end;

theorem Th62: :: CARD_1:62
for f being Function holds card f = card (dom f)
proof end;

registration
let X be finite set ;
cluster RelIncl X -> finite ;
coherence
RelIncl X is finite
proof end;
end;

:: from AMISTD_2, 2010.01.10, A.T
theorem :: CARD_1:63
for X being set st RelIncl X is finite holds
X is finite
proof end;

theorem :: CARD_1:64
for x being set
for k being Nat holds card (k --> x) = k
proof end;

begin

definition
let N, X be set ;
attr X is N -element means :Def7: :: CARD_1:def 7
 card X = N;
end;

:: deftheorem Def7 defines -element CARD_1:def 7 :
for N, X being set holds
( X is N -element iff card X = N );

registration
let N be Cardinal;
cluster N -element for set ;
existence
ex b1 being set st b1 is N -element
proof end;
end;

registration
cluster 0 -element -> empty for set ;
coherence
for b1 being set st b1 is 0 -element holds
b1 is empty
proof end;
cluster empty -> 0 -element for set ;
coherence
for b1 being set st b1 is empty holds
b1 is 0 -element
proof end;
end;

registration
let x be set ;
cluster {x} -> 1 -element ;
coherence
{x} is 1 -element
proof end;
end;

registration
let N be Cardinal;
cluster Relation-like Function-like N -element for set ;
existence
ex b1 being Function st b1 is N -element
proof end;
end;

registration
let N be Cardinal;
let f be N -element Function;
cluster dom f -> N -element ;
coherence
dom f is N -element
proof end;
end;

registration
cluster 1 -element -> non empty trivial for set ;
coherence
for b1 being set st b1 is 1 -element holds
( b1 is trivial & not b1 is empty )
proof end;
cluster non empty trivial -> 1 -element for set ;
coherence
for b1 being set st b1 is trivial & not b1 is empty holds
b1 is 1 -element
proof end;
end;

registration
let X be non empty set ;
cluster 1 -element for Element of bool X;
existence
ex b1 being Subset of X st b1 is 1 -element
proof end;
end;

definition
let X be non empty set ;
mode Singleton of X is 1 -element Subset of X;
end;

theorem :: CARD_1:65
for X being non empty set
for A being Singleton of X ex x being Element of X st A = {x}
proof end;

theorem Th66: :: CARD_1:66
for X, Y being set holds
( card X c= card Y iff ex f being Function st X c= f .: Y )
proof end;

theorem :: CARD_1:67
for X being set
for f being Function holds card (f .: X) c= card X by Th66;

theorem :: CARD_1:68
for X, Y being set st card X in card Y holds
Y \ X <> {}
proof end;

theorem :: CARD_1:69
for X, x being set holds
( X,[:X,{x}:] are_equipotent & card X = card [:X,{x}:] )
proof end;

:: from POLYFORM, 2011.07.25, A.T.
theorem :: CARD_1:70
for f being Function st f is one-to-one holds
card (dom f) = card (rng f)
proof end;