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Theorem List for Metamath Proof Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremac6s4 8001* Generalization of the Axiom of Choice to proper classes.  B is a collection  B ( x ) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )
 )
 
Theoremac6s5 8002* Generalization of the Axiom of Choice to proper classes.  B is a collection  B ( x ) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f A. x  e.  A  ( f `  x )  e.  B )
 
Theoremac8 8003* An Axiom of Choice equivalent. Given a family  x of mutually disjoint nonempty sets, there exists a set  y containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
 |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y ) )
 
Theoremac9s 8004* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes  B ( x ) (achieved via the Collection Principle cp 7445). (Contributed by NM, 29-Sep-2006.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
 
3.2.2  AC equivalents: well ordering, Zorn's lemma
 
Theoremnumthcor 8005* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
 |-  ( A  e.  V  ->  E. x  e.  On  A  ~<  x )
 
Theoremweth 8006* Well-ordering theorem: any set  A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)
 |-  ( A  e.  V  ->  E. x  x  We  A )
 
Theoremzorn2lem1 8007* Lemma for zorn2 8017. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) )  ->  ( F `  x )  e.  D )
 
Theoremzorn2lem2 8008* Lemma for zorn2 8017. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) )  ->  ( y  e.  x  ->  ( F `  y
 ) R ( F `
  x ) ) )
 
Theoremzorn2lem3 8009* Lemma for zorn2 8017. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( R  Po  A  /\  ( x  e.  On  /\  ( w  We  A  /\  D  =/= 
 (/) ) ) ) 
 ->  ( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
 
Theoremzorn2lem4 8010* Lemma for zorn2 8017. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   =>    |-  ( ( R  Po  A  /\  w  We  A )  ->  E. x  e.  On  D  =  (/) )
 
Theoremzorn2lem5 8011* Lemma for zorn2 8017. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  ->  ( F " x )  C_  A )
 
Theoremzorn2lem6 8012* Lemma for zorn2 8017. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( R  Po  A  ->  ( ( ( w  We  A  /\  x  e.  On )  /\  A. y  e.  x  H  =/=  (/) )  ->  R  Or  ( F " x ) ) )
 
Theoremzorn2lem7 8013* Lemma for zorn2 8017. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  F  = recs ( ( f  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u w v ) ) )   &    |-  C  =  {
 z  e.  A  |  A. g  e.  ran  f  g R z }   &    |-  D  =  { z  e.  A  |  A. g  e.  ( F " x ) g R z }   &    |-  H  =  { z  e.  A  |  A. g  e.  ( F " y ) g R z }   =>    |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. s
 ( ( s  C_  A  /\  R  Or  s
 )  ->  E. a  e.  A  A. r  e.  s  ( r R a  \/  r  =  a ) ) ) 
 ->  E. a  e.  A  A. b  e.  A  -.  a R b )
 
Theoremzorn2g 8014* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 8017 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. w ( ( w  C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x )
 ) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremzorng 8015* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 8018 avoids the Axiom of Choice by assuming that  A is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  A. z ( ( z  C_  A  /\ [ C.]  Or  z ) 
 ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzornn0g 8016* Variant of Zorn's lemma zorng 8015 in which  (/), the union of the empty chain, is not required to be an element of  A. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  ( ( A  e.  dom  card  /\  A  =/=  (/)  /\  A. z ( ( z 
 C_  A  /\  z  =/= 
 (/)  /\ [ C.]  Or  z
 )  ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzorn2 8017* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set  A (with an ordering relation  R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 8007 through zorn2lem7 8013; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8013. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( R  Po  A  /\  A. w ( ( w 
 C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x ) ) ) 
 ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremzorn 8018* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 8017 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
 |-  A  e.  _V   =>    |-  ( A. z
 ( ( z  C_  A  /\ [ C.]  Or  z
 )  ->  U. z  e.  A )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremzornn0 8019* Variant of Zorn's lemma zorn 8018 in which  (/), the union of the empty chain, is not required to be an element of  A. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A. z ( ( z  C_  A  /\  z  =/=  (/)  /\ [ C.]  Or  z )  ->  U. z  e.  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremttukeylem1 8020* Lemma for ttukey 8029. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   =>    |-  ( ph  ->  ( C  e.  A  <->  ( ~P C  i^i  Fin )  C_  A ) )
 
Theoremttukeylem2 8021* Lemma for ttukey 8029. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   =>    |-  ( ( ph  /\  ( C  e.  A  /\  D  C_  C ) ) 
 ->  D  e.  A )
 
Theoremttukeylem3 8022* Lemma for ttukey 8029. (Contributed by Mario Carneiro, 11-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  C  e.  On )  ->  ( G `  C )  =  if ( C  =  U. C ,  if ( C  =  (/) ,  B ,  U. ( G " C ) ) ,  ( ( G `  U. C )  u.  if ( ( ( G `
  U. C )  u. 
 { ( F `  U. C ) } )  e.  A ,  { ( F `  U. C ) } ,  (/) ) ) ) )
 
Theoremttukeylem4 8023* Lemma for ttukey 8029. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ph  ->  ( G `  (/) )  =  B )
 
Theoremttukeylem5 8024* Lemma for ttukey 8029. The  G function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  ( C  e.  On  /\  D  e.  On  /\  C  C_  D ) )  ->  ( G `  C ) 
 C_  ( G `  D ) )
 
Theoremttukeylem6 8025* Lemma for ttukey 8029. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ( ph  /\  C  e.  suc  ( card `  ( U. A  \  B ) ) )  ->  ( G `  C )  e.  A )
 
Theoremttukeylem7 8026* Lemma for ttukey 8029. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ph  ->  F : ( card `  ( U. A  \  B ) ) -1-1-onto-> ( U. A  \  B ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )   &    |-  G  = recs ( (
 z  e.  _V  |->  if ( dom  z  = 
 U. dom  z ,  if ( dom  z  =  (/) ,  B ,  U. ran  z ) ,  (
 ( z `  U. dom  z )  u.  if ( ( ( z `
  U. dom  z )  u.  { ( F `
  U. dom  z ) } )  e.  A ,  { ( F `  U.
 dom  z ) } ,  (/) ) ) ) ) )   =>    |-  ( ph  ->  E. x  e.  A  ( B  C_  x  /\  A. y  e.  A  -.  x  C.  y ) )
 
Theoremttukey2g 8027* The Teichmüller-Tukey Lemma ttukey 8029 with a slightly stronger conclusion: we can set up the maximal element of  A so that it also contains some given  B  e.  A as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( U. A  e.  dom  card  /\  B  e.  A  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  ( B  C_  x  /\  A. y  e.  A  -.  x  C.  y ) )
 
Theoremttukeyg 8028* The Teichmüller-Tukey Lemma ttukey 8029 stated with the "choice" as an antecedent (the hypothesis  U. A  e.  dom  card says that  U. A is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( U. A  e.  dom  card  /\  A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremttukey 8029* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If  A is a nonempty collection of finite character, then  A has a maximal element with respect to inclusion. Here "finite character" means that  x  e.  A iff every finite subset of  x is in  A. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A ) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
 
Theoremaxdclem 8030* Lemma for axdc 8032. (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  F  =  ( rec ( ( y  e. 
 _V  |->  ( g `  { z  |  y x z } )
 ) ,  s )  |`  om )   =>    |-  ( ( A. y  e.  ~P  dom  x (
 y  =/=  (/)  ->  (
 g `  y )  e.  y )  /\  ran  x 
 C_  dom  x  /\  E. z ( F `  K ) x z )  ->  ( K  e.  om  ->  ( F `  K ) x ( F `  suc  K ) ) )
 
Theoremaxdclem2 8031* Lemma for axdc 8032. Using the full Axiom of Choice, we can construct a choice function  g on  ~P dom  x. From this, we can build a sequence  F starting at any value  s  e.  dom  x by repeatedly applying  g to the set  ( F `  x ) (where  x is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  F  =  ( rec ( ( y  e. 
 _V  |->  ( g `  { z  |  y x z } )
 ) ,  s )  |`  om )   =>    |-  ( E. z  s x z  ->  ( ran  x  C_  dom  x  ->  E. f A. n  e. 
 om  ( f `  n ) x ( f `  suc  n ) ) )
 
Theoremaxdc 8032* This theorem derives ax-dc 7956 using ax-ac 7969 and ax-inf 7223. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (Contributed by Mario Carneiro, 25-Jan-2013.)
 |-  ( ( E. y E. z  y x z  /\  ran  x  C_  dom  x )  ->  E. f A. n  e.  om  ( f `  n ) x ( f `  suc  n ) )
 
Theoremfodom 8033 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 7985. AC is not needed for finite sets - see fodomfi 7020. See also fodomnum 7568. (Contributed by NM, 23-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( F : A -onto-> B  ->  B  ~<_  A )
 
Theoremfodomg 8034 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  ~<_  A ) )
 
Theoremfodomb 8035* Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
 |-  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B 
 ~<_  A ) )
 
Theoremwdomac 8036 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  ( X  ~<_*  Y  <->  X  ~<_  Y )
 
Theorembrdom3 8037* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x E* y  x f y  /\  A. x  e.  A  E. y  e.  B  y f x ) )
 
Theorembrdom5 8038* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y  x f y  /\  A. x  e.  A  E. y  e.  B  y
 f x ) )
 
Theorembrdom4 8039* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.)
 |-  B  e.  _V   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y ( y  e.  A  /\  x f y )  /\  A. x  e.  A  E. y  e.  B  y f x ) )
 
Theorembrdom7disj 8040* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( A  i^i  B )  =  (/)   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y ( y  e.  A  /\  { x ,  y }  e.  f
 )  /\  A. x  e.  A  E. y  e.  B  { y ,  x }  e.  f
 ) )
 
Theorembrdom6disj 8041* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( A  i^i  B )  =  (/)   =>    |-  ( A  ~<_  B  <->  E. f ( A. x  e.  B  E* y { x ,  y }  e.  f  /\  A. x  e.  A  E. y  e.  B  { y ,  x }  e.  f
 ) )
 
Theoremfin71ac 8042 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
 |- FinVII  = 
 Fin
 
Theoremimadomg 8043 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
 |-  ( A  e.  B  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
 
Theoremfnrndomg 8044 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
 |-  ( A  e.  B  ->  ( F  Fn  A  ->  ran  F  ~<_  A ) )
 
Theoremiunfo 8045* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   =>    |-  ( 2nd  |`  T ) : T -onto-> U_ x  e.  A  B
 
Theoremiundom2g 8046* An upper bound for the cardinality of an disjoint indexed union, with explicit choice principles. 
B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   &    |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )   &    |-  ( ph  ->  A. x  e.  A  B  ~<_  C )   =>    |-  ( ph  ->  T  ~<_  ( A  X.  C ) )
 
Theoremiundomg 8047* An upper bound for the cardinality of an indexed union, with explicit choice principles.  B depends on  x and should be thought of as  B ( x ). (Contributed by Mario Carneiro, 1-Sep-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  B )   &    |-  ( ph  ->  U_ x  e.  A  ( C  ^m  B )  e. AC  A )   &    |-  ( ph  ->  A. x  e.  A  B  ~<_  C )   &    |-  ( ph  ->  ( A  X.  C )  e. AC  U_ x  e.  A  B )   =>    |-  ( ph  ->  U_ x  e.  A  B  ~<_  ( A  X.  C ) )
 
Theoremiundom 8048* An upper bound for the cardinality of an indexed union.  C depends on  x and should be thought of as  C ( x ). (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B ) 
 ->  U_ x  e.  A  C 
 ~<_  ( A  X.  B ) )
 
Theoremunidom 8049* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
 |-  ( ( A  e.  V  /\  A. x  e.  A  x  ~<_  B ) 
 ->  U. A  ~<_  ( A  X.  B ) )
 
Theoremuniimadom 8050* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
 
Theoremuniimadomf 8051* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 8050 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
 |-  F/_ x F   &    |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
 
3.2.3  Cardinal number theorems using Axiom of Choice
 
Theoremcardval 8052* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 7508 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( card `  A )  =  |^| { x  e.  On  |  x  ~~  A }
 
Theoremcardid 8053 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  A  e.  _V   =>    |-  ( card `  A )  ~~  A
 
Theoremcardf 8054 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |- 
 card : _V --> On
 
Theoremcarden 8055 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank (see karden 7449). (Contributed by NM, 22-Oct-2003.)

 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( card `  A )  =  (
 card `  B )  <->  A  ~~  B ) )
 
Theoremcardeq0 8056 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  ( ( card `  A )  =  (/)  <->  A  =  (/) ) )
 
Theoremunsnen 8057 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( -.  B  e.  A  ->  ( A  u.  { B } )  ~~  suc  ( card `  A )
 )
 
Theoremcarddom 8058 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A )  C_  ( card `  B )  <->  A  ~<_  B )
 )
 
Theoremcardsdom 8059 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( card `  A )  e.  ( card `  B )  <->  A  ~<  B ) )
 
Theoremdomtri 8060 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theorementric 8061 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<  B  \/  A  ~~  B  \/  B  ~<  A )
 )
 
Theorementri2 8062 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  \/  B  ~<  A ) )
 
Theorementri3 8063 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  B  \/  B 
 ~<_  A ) )
 
Theoremsdomsdomcard 8064 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  ~<  B  <->  A  ~<  ( card `  B ) )
 
Theoremcanth3 8065 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how largei a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)
 |-  ( A  e.  V  ->  ( card `  A )  e.  ( card `  ~P A ) )
 
Theoreminfxpidm 8066 The cross product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 7526. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( om  ~<_  A  ->  ( A  X.  A ) 
 ~~  A )
 
Theoremondomon 8067* The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 7143. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { x  e.  On  |  x  ~<_  A }  e.  On )
 
Theoremcardmin 8068* The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)
 |-  ( A  e.  V  ->  ( card `  |^| { x  e.  On  |  A  ~<  x } )  =  |^| { x  e.  On  |  A  ~<  x } )
 
Theoremficard 8069 A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen 2-Sep-2009.)
 |-  ( A  e.  V  ->  ( A  e.  Fin  <->  ( card `  A )  e. 
 om ) )
 
Theoremkonigthlem 8070* Lemma for konigth 8071. (Contributed by Mario Carneiro, 22-Feb-2013.)
 |-  A  e.  _V   &    |-  S  =  U_ i  e.  A  ( M `  i )   &    |-  P  =  X_ i  e.  A  ( N `  i )   &    |-  D  =  ( i  e.  A  |->  ( a  e.  ( M `
  i )  |->  ( ( f `  a
 ) `  i )
 ) )   &    |-  E  =  ( i  e.  A  |->  ( e `  i ) )   =>    |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `
  i )  ->  S  ~<  P )
 
Theoremkonigth 8071* Konig's Theorem. If  m ( i ) 
~<  n ( i ) for all 
i  e.  A, then  sum_ i  e.  A m ( i )  ~<  prod_ i  e.  A n ( i ), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting  m ( i )  =  (/), this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)
 |-  A  e.  _V   &    |-  S  =  U_ i  e.  A  ( M `  i )   &    |-  P  =  X_ i  e.  A  ( N `  i )   =>    |-  ( A. i  e.  A  ( M `  i )  ~<  ( N `
  i )  ->  S  ~<  P )
 
Theoremalephsucpw 8072 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8182 or gchaleph2 8178.) (Contributed by NM, 27-Aug-2005.)
 |-  ( aleph `  suc  A )  ~<_  ~P ( aleph `  A )
 
Theoremaleph1 8073 The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.)
 |-  ( aleph `  1o )  ~<_  ( 2o  ^m  ( aleph `  (/) ) )
 
Theoremalephval2 8074* An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.)
 |-  ( ( A  e.  On  /\  (/)  e.  A ) 
 ->  ( aleph `  A )  =  |^| { x  e. 
 On  |  A. y  e.  A  ( aleph `  y
 )  ~<  x } )
 
Theoremdominfac 8075 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 7969. See dominf 7955 for a version proved from ax-cc 7945. (Contributed by NM, 25-Mar-2007.)
 |-  A  e.  _V   =>    |-  ( ( A  =/=  (/)  /\  A  C_  U. A )  ->  om  ~<_  A )
 
3.2.4  Cardinal number arithmetic using Axiom of Choice
 
Theoremiunctb 8076* The countable union of countable sets is countable (indexed union version of unictb 8077). (Contributed by Mario Carneiro, 18-Jan-2014.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  B  ~<_  om )  ->  U_ x  e.  A  B  ~<_  om )
 
Theoremunictb 8077* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 8076 for indexed union version. (Contributed by NM, 26-Mar-2006.)
 |-  ( ( A  ~<_  om  /\  A. x  e.  A  x  ~<_  om )  ->  U. A  ~<_  om )
 
Theoreminfmap 8078* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. (Contributed by NM, 1-Oct-2004.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B 
 ~<_  A )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
 C_  A  /\  x  ~~  B ) } )
 
Theoremalephadd 8079 The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( aleph `  A )  +c  ( aleph `  B ) )  ~~  ( (
 aleph `  A )  u.  ( aleph `  B )
 )
 
Theoremalephmul 8080 The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( aleph `  A )  X.  ( aleph `  B ) )  ~~  ( ( aleph `  A )  u.  ( aleph `  B ) ) )
 
Theoremalephexp1 8081 An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B )  ->  (
 ( aleph `  A )  ^m  ( aleph `  B )
 )  ~~  ( 2o  ^m  ( aleph `  B )
 ) )
 
Theoremalephsuc3 8082* An alternate representation of a successor aleph. Compare alephsuc 7579 and alephsuc2 7591. Equality can be obtained by taking the  card of the right-hand side then using alephcard 7581 and carden 8055. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  e.  On  ->  ( aleph `  suc  A ) 
 ~~  { x  e.  On  |  x  ~~  ( aleph `  A ) } )
 
Theoremalephexp2 8083* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8081 (which works if the base is less than or equal to the exponent) and infmap 8078 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
 C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) }
 )
 
3.2.5  Cofinality using Axiom of Choice
 
Theoremalephreg 8084 A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  ( cf `  ( aleph `  suc  A ) )  =  ( aleph `  suc  A )
 
Theorempwcfsdom 8085* A corollary of Konig's Theorem konigth 8071. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  H  =  ( y  e.  ( cf `  ( aleph `  A ) )  |->  (har `  ( f `  y
 ) ) )   =>    |-  ( aleph `  A )  ~<  ( ( aleph `  A )  ^m  ( cf `  ( aleph `  A ) ) )
 
Theoremcfpwsdom 8086 A corollary of Konig's Theorem konigth 8071. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
 |-  B  e.  _V   =>    |-  ( 2o  ~<_  B  ->  (
 aleph `  A )  ~<  (
 cf `  ( card `  ( B  ^m  ( aleph `  A ) ) ) ) )
 
Theoremalephom 8087 From canth2 6899, we know that  (
aleph `  0 )  < 
( 2 ^ om ), but we cannot prove that  ( 2 ^ om )  =  ( aleph `  1 ) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement  ( aleph `  A )  <  ( 2 ^ om ) is consistent for any ordinal  A). However, we can prove that  ( 2 ^ om ) is not equal to  ( aleph `  om ), nor  ( aleph `  ( aleph `  om ) ), on cofinality grounds, because by Konig's Theorem konigth 8071 (in the form of cfpwsdom 8086), 
( 2 ^ om ) has uncountable cofinality, which eliminates limit alephs like 
( aleph `  om ). (The first limit aleph that is not eliminated is  (
aleph `  ( aleph `  1
) ), which has cofinality  ( aleph `  1 ).) (Contributed by Mario Carneiro, 21-Mar-2013.)
 |-  ( card `  ( 2o  ^m 
 om ) )  =/=  ( aleph `  om )
 
Theoremsmobeth 8088 The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as  ( card `  ( R1 `  ( om  +o  A ) ) ), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)
 |- 
 Smo  ( card  o.  R1 )
 
3.3  ZFC Axioms with no distinct variable requirements
 
Theoremnd1 8089 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  y  e.  z
 )
 
Theoremnd2 8090 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. x  z  e.  y
 )
 
Theoremnd3 8091 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  x  e.  y
 )
 
Theoremnd4 8092 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
 |-  ( A. x  x  =  y  ->  -.  A. z  y  e.  x )
 
Theoremaxextnd 8093 A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
 |- 
 E. x ( ( x  e.  y  <->  x  e.  z
 )  ->  y  =  z )
 
Theoremaxrepndlem1 8094* Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. y  y  =  z  ->  E. x ( E. y A. z ( ph  ->  z  =  y )  ->  A. z ( z  e.  x  <->  E. x ( x  e.  y  /\  A. y ph ) ) ) )
 
Theoremaxrepndlem2 8095 Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( ( ( -. 
 A. x  x  =  y  /\  -.  A. x  x  =  z
 )  /\  -.  A. y  y  =  z )  ->  E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( z  e.  x  <->  E. x ( x  e.  y  /\  A. y ph ) ) ) )
 
Theoremaxrepnd 8096 A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |- 
 E. x ( E. y A. z ( ph  ->  z  =  y ) 
 ->  A. z ( A. y  z  e.  x  <->  E. x ( A. z  x  e.  y  /\  A. y ph ) ) )
 
Theoremaxunndlem1 8097* Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |- 
 E. x A. y
 ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x )
 
Theoremaxunnd 8098 A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
 |- 
 E. x A. y
 ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x )
 
Theoremaxpowndlem1 8099 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
 |-  ( A. x  x  =  y  ->  ( -.  x  =  y  ->  E. x A. y
 ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) )
 
Theoremaxpowndlem2 8100* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( -.  x  =  y  ->  E. x A. y
 ( A. x ( E. z  x  e.  y  ->  A. y  x  e.  z )  ->  y  e.  x ) ) ) )
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