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Theorem List for Metamath Proof Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcardsucinf 7501 The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( card `  suc  A )  =  ( card `  A ) )
 
Theoremcardsucnn 7502 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 7501. (Contributed by NM, 7-Nov-2008.)
 |-  ( A  e.  om  ->  ( card `  suc  A )  =  suc  ( card `  A ) )
 
Theoremcardom 7503 The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.)
 |-  ( card `  om )  = 
 om
 
Theoremcarden2 7504 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 8055, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( ( card `  A )  =  ( card `  B )  <->  A 
 ~~  B ) )
 
Theoremcardsdom2 7505 A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( ( card `  A )  e.  ( card `  B )  <->  A 
 ~<  B ) )
 
Theoremdomtri2 7506 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  e.  dom  card
 )  ->  ( A  ~<_  B 
 <->  -.  B  ~<  A ) )
 
Theoremnnsdomel 7507 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B 
 <->  A  ~<  B )
 )
 
Theoremcardval2 7508* An alternate version of the value of the cardinal number of a set. Compare cardval 8052. This theorem could be used to give us a simpler definition of  card in place of df-card 7456. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
 |-  ( A  e.  dom  card 
 ->  ( card `  A )  =  { x  e.  On  |  x  ~<  A }
 )
 
Theoremisinffi 7509* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 6961 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  E. f  f : B -1-1-> A )
 
Theoremfidomtri 7510 Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  V ) 
 ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theoremfidomtri2 7511 Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.)
 |-  ( ( A  e.  V  /\  B  e.  Fin )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theoremharsdom 7512 The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  dom  card 
 ->  A  ~<  (har `  A ) )
 
Theoremonsdom 7513* Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  dom  card 
 ->  E. x  e.  On  A  ~<  x )
 
Theoremharval2 7514* An alternative expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  dom  card 
 ->  (har `  A )  =  |^| { x  e. 
 On  |  A  ~<  x } )
 
Theoremcardmin2 7515* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  ( E. x  e. 
 On  A  ~<  x  <->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
 
Theorempm54.43lem 7516* In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7485), so that their  A  e.  1 means, in our notation,  A  e.  {
x  |  ( card `  x )  =  1o }. Here we show that this is equivalent to  A  ~~  1o so that we can use the latter more convenient notation in pm54.43 7517. (Contributed by NM, 4-Nov-2013.)
 |-  ( A  ~~  1o  <->  A  e.  { x  |  (
 card `  x )  =  1o } )
 
Theorempm54.43 7517 Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2.

Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 7485), so that their  A  e.  1 means, in our notation,  A  e.  { x  |  (
card `  x )  =  1o } which is the same as  A  ~~  1o by pm54.43lem 7516. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.)

Theorem pm110.643 7687 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.)

 |-  ( ( A  ~~  1o  /\  B  ~~  1o )  ->  ( ( A  i^i  B )  =  (/) 
 <->  ( A  u.  B )  ~~  2o ) )
 
Theorempr2nelem 7518 Lemma for pr2ne 7519. (Contributed by FL, 17-Aug-2008.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B ) 
 ->  { A ,  B }  ~~  2o )
 
Theorempr2ne 7519 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { A ,  B }  ~~  2o  <->  A  =/=  B ) )
 
Theoremprdom2 7520 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A ,  B }  ~<_  2o )
 
Theoremen2eqpr 7521 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C ) 
 ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
 
Theoremdif1card 7522 The cardinality of a non-empty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)
 |-  ( ( A  e.  Fin  /\  X  e.  A ) 
 ->  ( card `  A )  =  suc  ( card `  ( A  \  { X }
 ) ) )
 
Theoremleweon 7523* Lexicographical order is a well-ordering of  On 
X.  On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 7524, this order is not set-like, as the preimage of  <. 1o ,  (/) >. is the proper class  ( { (/) }  X.  On ). (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  L  =  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X. 
 On ) )  /\  ( ( 1st `  x )  e.  ( 1st `  y )  \/  (
 ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) ) }   =>    |-  L  We  ( On  X.  On )
 
Theoremr0weon 7524* A set-like well ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  L  =  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X. 
 On ) )  /\  ( ( 1st `  x )  e.  ( 1st `  y )  \/  (
 ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) ) }   &    |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  (
 ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }   =>    |-  ( R  We  ( On  X.  On )  /\  R Se  ( On  X.  On ) )
 
Theoreminfxpenlem 7525* Lemma for infxpen 7526. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  L  =  { <. x ,  y >.  |  ( ( x  e.  ( On  X.  On )  /\  y  e.  ( On  X. 
 On ) )  /\  ( ( 1st `  x )  e.  ( 1st `  y )  \/  (
 ( 1st `  x )  =  ( 1st `  y
 )  /\  ( 2nd `  x )  e.  ( 2nd `  y ) ) ) ) }   &    |-  R  =  { <. z ,  w >.  |  ( ( z  e.  ( On  X.  On )  /\  w  e.  ( On  X.  On ) )  /\  ( ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  e.  ( ( 1st `  w )  u.  ( 2nd `  w ) )  \/  (
 ( ( 1st `  z
 )  u.  ( 2nd `  z ) )  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )  /\  z L w ) ) ) }   &    |-  Q  =  ( R  i^i  ( ( a  X.  a )  X.  ( a  X.  a ) ) )   &    |-  ( ph  <->  ( ( a  e.  On  /\  A. m  e.  a  ( om  C_  m  ->  ( m  X.  m )  ~~  m ) )  /\  ( om  C_  a  /\  A. m  e.  a  m 
 ~<  a ) ) )   &    |-  M  =  ( ( 1st `  w )  u.  ( 2nd `  w ) )   &    |-  J  = OrdIso ( Q ,  ( a  X.  a ) )   =>    |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
 
Theoreminfxpen 7526 Every infinite ordinal is equinumerous to its cross product. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation  R is a well-ordering of  ( On  X.  On ) with the additional property that  R-initial segments of  ( x  X.  x ) (where  x is a limit ordinal) are of cardinality at most  x. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  ( A  X.  A )  ~~  A )
 
Theoremxpomen 7527 The cross product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( om  X.  om )  ~~  om
 
Theoreminfxpidm2 7528 The cross product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 8066. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A ) 
 ->  ( A  X.  A )  ~~  A )
 
Theoreminfxpenc 7529* A canonical version of infxpen 7526, by a completely different approach (although it uses infxpen 7526 via xpomen 7527). Using Cantor's normal form, we can show that  A  ^o  B respects equinumerosity (oef1o 7285), so that all the steps of  ( om ^ W
)  x.  ( om
^ W )  ~~  om
^ ( 2 W )  ~~  ( om ^
2 ) ^ W  ~~  om ^ W can be verified using bijections to do the ordinal commutations. (The assumption on  N can be satisfied using cnfcom3c 7293.) (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  om  C_  A )   &    |-  ( ph  ->  W  e.  ( On  \  1o ) )   &    |-  ( ph  ->  F : ( om  ^o  2o ) -1-1-onto-> om )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  ( ph  ->  N : A -1-1-onto-> ( om  ^o  W ) )   &    |-  K  =  ( y  e.  { x  e.  ( ( om  ^o  2o )  ^m  W )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( F  o.  (
 y  o.  `' (  _I  |`  W ) ) ) )   &    |-  H  =  ( ( ( om CNF  W )  o.  K )  o.  `' ( ( om  ^o  2o ) CNF  W )
 )   &    |-  L  =  ( y  e.  { x  e.  ( om  ^m  ( W  .o  2o ) )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( (  _I  |`  om )  o.  ( y  o.  `' ( Y  o.  `' X ) ) ) )   &    |-  X  =  ( z  e.  2o ,  w  e.  W  |->  ( ( W  .o  z )  +o  w ) )   &    |-  Y  =  ( z  e.  2o ,  w  e.  W  |->  ( ( 2o  .o  w )  +o  z
 ) )   &    |-  J  =  ( ( ( om CNF  ( 2o  .o  W ) )  o.  L )  o.  `' ( om CNF  ( W  .o  2o ) ) )   &    |-  Z  =  ( x  e.  ( om  ^o  W ) ,  y  e.  ( om  ^o  W ) 
 |->  ( ( ( om  ^o  W )  .o  x )  +o  y ) )   &    |-  T  =  ( x  e.  A ,  y  e.  A  |->  <. ( N `  x ) ,  ( N `  y ) >. )   &    |-  G  =  ( `' N  o.  ( ( ( H  o.  J )  o.  Z )  o.  T ) )   =>    |-  ( ph  ->  G : ( A  X.  A ) -1-1-onto-> A )
 
Theoreminfxpenc2lem1 7530* Lemma for infxpenc2 7533. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( n `
  b ) : b -1-1-onto-> ( om  ^o  w ) ) )   &    |-  W  =  ( `' ( x  e.  ( On  \  1o )  |->  ( om  ^o  x ) ) `  ran  (  n `  b
 ) )   =>    |-  ( ( ph  /\  (
 b  e.  A  /\  om  C_  b ) )  ->  ( W  e.  ( On  \  1o )  /\  ( n `  b ) : b -1-1-onto-> ( om  ^o  W ) ) )
 
Theoreminfxpenc2lem2 7531* Lemma for infxpenc2 7533. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( n `
  b ) : b -1-1-onto-> ( om  ^o  w ) ) )   &    |-  W  =  ( `' ( x  e.  ( On  \  1o )  |->  ( om  ^o  x ) ) `  ran  (  n `  b
 ) )   &    |-  ( ph  ->  F : ( om  ^o  2o ) -1-1-onto-> om )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   &    |-  K  =  ( y  e.  { x  e.  ( ( om  ^o  2o )  ^m  W )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( F  o.  (
 y  o.  `' (  _I  |`  W ) ) ) )   &    |-  H  =  ( ( ( om CNF  W )  o.  K )  o.  `' ( ( om  ^o  2o ) CNF  W )
 )   &    |-  L  =  ( y  e.  { x  e.  ( om  ^m  ( W  .o  2o ) )  |  ( `' x " ( _V  \  1o ) )  e.  Fin } 
 |->  ( (  _I  |`  om )  o.  ( y  o.  `' ( Y  o.  `' X ) ) ) )   &    |-  X  =  ( z  e.  2o ,  w  e.  W  |->  ( ( W  .o  z )  +o  w ) )   &    |-  Y  =  ( z  e.  2o ,  w  e.  W  |->  ( ( 2o  .o  w )  +o  z
 ) )   &    |-  J  =  ( ( ( om CNF  ( 2o  .o  W ) )  o.  L )  o.  `' ( om CNF  ( W  .o  2o ) ) )   &    |-  Z  =  ( x  e.  ( om  ^o  W ) ,  y  e.  ( om  ^o  W ) 
 |->  ( ( ( om  ^o  W )  .o  x )  +o  y ) )   &    |-  T  =  ( x  e.  b ,  y  e.  b  |->  <. ( ( n `
  b ) `  x ) ,  (
 ( n `  b
 ) `  y ) >. )   &    |-  G  =  ( `' ( n `  b
 )  o.  ( ( ( H  o.  J )  o.  Z )  o.  T ) )   =>    |-  ( ph  ->  E. g A. b  e.  A  ( om  C_  b  ->  ( g `  b
 ) : ( b  X.  b ) -1-1-onto-> b ) )
 
Theoreminfxpenc2lem3 7532* Lemma for infxpenc2 7533. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  A  e.  On )   &    |-  ( ph  ->  A. b  e.  A  ( om  C_  b  ->  E. w  e.  ( On  \  1o ) ( n `
  b ) : b -1-1-onto-> ( om  ^o  w ) ) )   &    |-  W  =  ( `' ( x  e.  ( On  \  1o )  |->  ( om  ^o  x ) ) `  ran  (  n `  b
 ) )   &    |-  ( ph  ->  F : ( om  ^o  2o ) -1-1-onto-> om )   &    |-  ( ph  ->  ( F `  (/) )  =  (/) )   =>    |-  ( ph  ->  E. g A. b  e.  A  ( om  C_  b  ->  ( g `  b ) : ( b  X.  b ) -1-1-onto-> b ) )
 
Theoreminfxpenc2 7533* Existence form of infxpenc 7529. A "uniform" or "canonical" version of infxpen 7526, asserting the existence of a single function  g that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( A  e.  On  ->  E. g A. b  e.  A  ( om  C_  b  ->  ( g `  b
 ) : ( b  X.  b ) -1-1-onto-> b ) )
 
Theoremiunmapdisj 7534* The union  U_ n  e.  C ( A  ^m  n ) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.)
 |- 
 E* n ( n  e.  C  /\  B  e.  ( A  ^m  n ) )
 
Theoremfseqenlem1 7535* Lemma for fseqen 7538. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  F : ( A  X.  A ) -1-1-onto-> A )   &    |-  G  = seq𝜔 ( ( n  e. 
 _V ,  f  e. 
 _V  |->  ( x  e.  ( A  ^m  suc  n )  |->  ( ( f `
  ( x  |`  n ) ) F ( x `  n ) ) ) ) ,  { <. (/) ,  B >. } )   =>    |-  ( ( ph  /\  C  e.  om )  ->  ( G `  C ) : ( A  ^m  C ) -1-1-> A )
 
Theoremfseqenlem2 7536* Lemma for fseqen 7538. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  F : ( A  X.  A ) -1-1-onto-> A )   &    |-  G  = seq𝜔 ( ( n  e. 
 _V ,  f  e. 
 _V  |->  ( x  e.  ( A  ^m  suc  n )  |->  ( ( f `
  ( x  |`  n ) ) F ( x `  n ) ) ) ) ,  { <. (/) ,  B >. } )   &    |-  K  =  ( y  e.  U_ k  e.  om  ( A  ^m  k )  |->  <. dom  y ,  ( ( G `  dom  y ) `  y
 ) >. )   =>    |-  ( ph  ->  K : U_ k  e.  om  ( A  ^m  k )
 -1-1-> ( om  X.  A ) )
 
Theoremfseqdom 7537* One half of fseqen 7538. (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( A  e.  V  ->  ( om  X.  A ) 
 ~<_  U_ n  e.  om  ( A  ^m  n ) )
 
Theoremfseqen 7538* A set that is equinumerous to its cross product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.)
 |-  ( ( ( A  X.  A )  ~~  A  /\  A  =/=  (/) )  ->  U_ n  e.  om  ( A  ^m  n )  ~~  ( om  X.  A ) )
 
Theoreminfpwfidom 7539 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
 
Theoremdfac8alem 7540* Lemma for dfac8a 7541. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
 |-  F  = recs ( G )   &    |-  G  =  ( f  e.  _V  |->  ( g `  ( A 
 \  ran  f )
 ) )   =>    |-  ( A  e.  C  ->  ( E. g A. y  e.  ~P  A ( y  =/=  (/)  ->  (
 g `  y )  e.  y )  ->  A  e.  dom  card ) )
 
Theoremdfac8a 7541* Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  B  ->  ( E. h A. y  e.  ~P  A ( y  =/=  (/)  ->  ( h `  y )  e.  y )  ->  A  e.  dom  card ) )
 
Theoremdfac8b 7542* The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  dom  card 
 ->  E. x  x  We  A )
 
Theoremdfac8clem 7543* Lemma for dfac8c 7544. (Contributed by Mario Carneiro, 10-Jan-2013.)
 |-  F  =  ( s  e.  ( A  \  { (/) } )  |->  (
 iota_ a  e.  s A. b  e.  s  -.  b r a ) )   =>    |-  ( A  e.  B  ->  ( E. r  r  We  U. A  ->  E. f A. z  e.  A  ( z  =/=  (/)  ->  ( f `  z )  e.  z
 ) ) )
 
Theoremdfac8c 7544* If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  B  ->  ( E. r  r  We  U. A  ->  E. f A. z  e.  A  ( z  =/=  (/)  ->  ( f `  z )  e.  z
 ) ) )
 
Theoremac10ct 7545* A proof of the Well ordering theorem weth 8006, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( E. y  e. 
 On  A  ~<_  y  ->  E. x  x  We  A )
 
Theoremween 7546* A set is numerable iff it can be well ordered. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  dom  card  <->  E. r  r  We  A )
 
Theoremac5num 7547* A version of ac5b 7989 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( U. A  e.  dom  card  /\  -.  (/)  e.  A )  ->  E. f ( f : A --> U. A  /\  A. x  e.  A  ( f `  x )  e.  x )
 )
 
Theoremondomen 7548 If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( ( A  e.  On  /\  B  ~<_  A ) 
 ->  B  e.  dom  card )
 
Theoremnumdom 7549 A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  ~<_  A ) 
 ->  B  e.  dom  card )
 
Theoremssnum 7550 A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  B  C_  A )  ->  B  e.  dom  card
 )
 
Theoremonssnum 7551 All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)
 |-  ( ( A  e.  V  /\  A  C_  On )  ->  A  e.  dom  card
 )
 
Theoremindcardi 7552* Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  dom  card )   &    |-  (
 ( ph  /\  R  ~<_  T  /\  A. y ( S  ~<  R 
 ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
 
Theoremacnrcl 7553 Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  e. AC  A  ->  A  e.  _V )
 
Theoremacneq 7554 Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  =  C  -> AC  A  = AC  C )
 
Theoremisacn 7555* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  e.  W )  ->  ( X  e. AC  A  <->  A. f  e.  (
 ( ~P X  \  { (/) } )  ^m  A ) E. g A. x  e.  A  ( g `  x )  e.  ( f `  x ) ) )
 
Theoremacni 7556* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( X  e. AC  A 
 /\  F : A --> ( ~P X  \  { (/)
 } ) )  ->  E. g A. x  e.  A  ( g `  x )  e.  ( F `  x ) )
 
Theoremacni2 7557* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( X  e. AC  A 
 /\  A. x  e.  A  ( B  C_  X  /\  B  =/=  (/) ) )  ->  E. g ( g : A --> X  /\  A. x  e.  A  (
 g `  x )  e.  B ) )
 
Theoremacni3 7558* The property of being a choice set of length  A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( y  =  ( g `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( X  e. AC  A  /\  A. x  e.  A  E. y  e.  X  ph )  ->  E. g ( g : A --> X  /\  A. x  e.  A  ps ) )
 
Theoremacnlem 7559* Construct a mapping satisfying the consequent of isacn 7555. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  (
 f `  x )
 )  ->  E. g A. x  e.  A  ( g `  x )  e.  ( f `  x ) )
 
Theoremnumacn 7560 A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  V  ->  ( X  e.  dom  card 
 ->  X  e. AC  A ) )
 
Theoremfinacn 7561 Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  Fin  -> AC  A  =  _V )
 
Theoremacndom 7562 A set with long choice sequences also has shorter choice sequences, where "shorter" here means the new index set is dominated by the old index set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  ~<_  B  ->  ( X  e. AC  B  ->  X  e. AC  A ) )
 
Theoremacnnum 7563 A set  X which has choice sequences on it of length 
~P X is well-orderable (and hence has choice sequences of every length). (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  e. AC  ~P X  <->  X  e.  dom  card )
 
Theoremacnen 7564 The class of choice sets of length 
A is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  ~~  B  -> AC  A  = AC  B )
 
Theoremacndom2 7565 A set smaller than one with choice sequences of length  A also has choice sequences of length 
A. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  ~<_  Y  ->  ( Y  e. AC  A  ->  X  e. AC  A ) )
 
Theoremacnen2 7566 The class of sets with choice sequences of length  A is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( X  ~~  Y  ->  ( X  e. AC  A  <->  Y  e. AC  A ) )
 
Theoremfodomacn 7567 A version of fodom 8033 that doesn't require the Axiom of Choice ax-ac 7969. If  A has choice sequences of length  B, then any surjection from  A to  B can be inverted to an injection the other way. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e. AC  B  ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
 
Theoremfodomnum 7568 A version of fodom 8033 that doesn't require the Axiom of Choice ax-ac 7969. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( A  e.  dom  card 
 ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
 
Theoremfonum 7569 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  dom  card  /\  F : A -onto-> B )  ->  B  e.  dom  card )
 
Theoremnumwdom 7570 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( A  e.  dom  card  /\  B  ~<_*  A )  ->  B  e.  dom  card )
 
Theoremfodomfi2 7571 Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
 
Theoremwdomfil 7572 Weak dominance agrees with normal for finite left sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( X  e.  Fin  ->  ( X  ~<_*  Y  <->  X  ~<_  Y )
 )
 
Theoreminfpwfien 7573 Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A ) 
 ->  ( ~P A  i^i  Fin )  ~~  A )
 
Theoreminffien 7574 The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( ( A  e.  dom  card  /\  om  ~<_  A ) 
 ->  ( fi `  A )  ~~  A )
 
Theoremwdomnumr 7575 Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( B  e.  dom  card 
 ->  ( A  ~<_*  B  <->  A  ~<_  B )
 )
 
Theoremalephfnon 7576 The aleph function is a function on the class of ordinal numbers. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  aleph  Fn  On
 
Theoremaleph0 7577 The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers  om (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written 
aleph_0. Exercise 3 of [TakeutiZaring] p. 91. Also Definition 12(i) of [Suppes] p. 228. From Moshé Machover, Set Theory, Logic, and Their Limitations, p. 95: "Aleph...the first letter in the Hebrew alphabet...is also the first letter of the Hebrew word...(einsoph, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism." (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( aleph `  (/) )  = 
 om
 
Theoremalephlim 7578* Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( ( A  e.  V  /\  Lim  A )  ->  ( aleph `  A )  =  U_ x  e.  A  ( aleph `  x )
 )
 
Theoremalephsuc 7579 Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91. Here we express the successor aleph in terms of the Hartogs function df-har 7156, which gives the smallest ordinal that strictly dominates its argument (or the supremum of all ordinals that are dominated by the argument). (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  (har `  ( aleph `  A ) ) )
 
Theoremalephon 7580 An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  ( aleph `  A )  e.  On
 
Theoremalephcard 7581 Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( card `  ( aleph `  A ) )  =  ( aleph `  A )
 
Theoremalephnbtwn 7582 No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( ( card `  B )  =  B  ->  -.  ( ( aleph `  A )  e.  B  /\  B  e.  ( aleph ` 
 suc  A ) ) )
 
Theoremalephnbtwn2 7583 No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |- 
 -.  ( ( aleph `  A )  ~<  B  /\  B  ~<  ( aleph `  suc  A ) )
 
Theoremalephordilem1 7584 Lemma for alephordi 7585. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
 |-  ( A  e.  On  ->  ( aleph `  A )  ~<  ( aleph `  suc  A ) )
 
Theoremalephordi 7585 Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B )
 ) )
 
Theoremalephord 7586 Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B 
 <->  ( aleph `  A )  ~<  ( aleph `  B )
 ) )
 
Theoremalephord2 7587 Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B 
 <->  ( aleph `  A )  e.  ( aleph `  B )
 ) )
 
Theoremalephord2i 7588 Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.)
 |-  ( B  e.  On  ->  ( A  e.  B  ->  ( aleph `  A )  e.  ( aleph `  B )
 ) )
 
Theoremalephord3 7589 Ordering property of the aleph function. (Contributed by NM, 11-Nov-2003.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  ( aleph `  A )  C_  ( aleph `  B )
 ) )
 
Theoremalephsucdom 7590 A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( B  e.  On  ->  ( A  ~<_  ( aleph `  B )  <->  A  ~<  ( aleph ` 
 suc  B ) ) )
 
Theoremalephsuc2 7591* An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 7143 function by transfinite recursion, starting from 
om. Using this theorem we could define the aleph function with  { z  e.  On  |  z  ~<_  x } in place of  |^| { z  e.  On  |  x 
~<  z } in df-aleph 7457. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A )  =  { x  e. 
 On  |  x  ~<_  (
 aleph `  A ) }
 )
 
Theoremalephdom 7592 Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B 
 <->  ( aleph `  A )  ~<_  ( aleph `  B )
 ) )
 
Theoremalephgeom 7593 Every aleph is greater than or equal to the set of natural numbers. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  On  <->  om  C_  ( aleph `  A )
 )
 
Theoremalephislim 7594 Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  On  <->  Lim  ( aleph `  A )
 )
 
Theoremaleph11 7595 The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A )  =  (
 aleph `  B )  <->  A  =  B ) )
 
Theoremalephf1 7596 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. (Contributed by by Mario Carneiro, 2-Feb-2013.)
 |-  aleph : On -1-1-> On
 
Theoremalephsdom 7597 If an ordinal is smaller than an initial ordinal, it is strictly dominated by it. (Contributed by Jeff Hankins, 24-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  ( aleph `  B )  <->  A 
 ~<  ( aleph `  B )
 ) )
 
Theoremalephdom2 7598 A dominated initial ordinal is included. (Contributed by Jeff Hankins, 24-Oct-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A )  C_  B  <->  (
 aleph `  A )  ~<_  B ) )
 
Theoremalephle 7599 The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 7620, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.)
 |-  ( A  e.  On  ->  A  C_  ( aleph `  A ) )
 
Theoremcardaleph 7600* Given any transfinite cardinal number  A, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
 |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { x  e.  On  |  A  C_  ( aleph `  x ) } ) )
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