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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisfi 6901* Express " A is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose " Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
 |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x )
 
Theoremenssdom 6902 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~~  C_  ~<_
 
Theoremdfdom2 6903 Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
 |-  ~<_  =  (  ~<  u.  ~~  )
 
Theoremendom 6904 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
 |-  ( A  ~~  B  ->  A  ~<_  B )
 
Theoremsdomdom 6905 Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<  B  ->  A  ~<_  B )
 
Theoremsdomnen 6906 Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<  B  ->  -.  A  ~~  B )
 
Theorembrdom2 6907 Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
 |-  ( A  ~<_  B  <->  ( A  ~<  B  \/  A  ~~  B ) )
 
Theorembren2 6908 Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
 |-  ( A  ~~  B  <->  ( A  ~<_  B  /\  -.  A  ~<  B ) )
 
Theoremenrefg 6909 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  ->  A  ~~  A )
 
Theoremenref 6910 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  e.  _V   =>    |-  A  ~~  A
 
Theoremeqeng 6911 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
 |-  ( A  e.  V  ->  ( A  =  B  ->  A  ~~  B ) )
 
Theoremdomrefg 6912 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
 |-  ( A  e.  V  ->  A  ~<_  A )
 
Theoremen2d 6913* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( x  e.  A  ->  C  e.  _V ) )   &    |-  ( ph  ->  ( y  e.  B  ->  D  e.  _V ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D )
 ) )   =>    |-  ( ph  ->  A  ~~  B )
 
Theoremen3d 6914* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( y  e.  B  ->  D  e.  A ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <->  y  =  C ) ) )   =>    |-  ( ph  ->  A 
 ~~  B )
 
Theoremen2i 6915* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  e.  A  ->  C  e.  _V )   &    |-  ( y  e.  B  ->  D  e.  _V )   &    |-  ( ( x  e.  A  /\  y  =  C )  <->  ( y  e.  B  /\  x  =  D ) )   =>    |-  A  ~~  B
 
Theoremen3i 6916* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( y  e.  B  ->  D  e.  A )   &    |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  =  D  <->  y  =  C ) )   =>    |-  A  ~~  B
 
Theoremdom2lem 6917* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  C ) : A -1-1-> B )
 
Theoremdom2d 6918* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   =>    |-  ( ph  ->  ( B  e.  R  ->  A  ~<_  B ) )
 
Theoremdom3d 6919* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
 |-  ( ph  ->  ( x  e.  A  ->  C  e.  B ) )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y ) ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  A  ~<_  B )
 
Theoremdom2 6920* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
 |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y
 ) )   =>    |-  ( B  e.  V  ->  A  ~<_  B )
 
Theoremdom3 6921* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain.  C and  D can be read  C ( x ) and  D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
 |-  ( x  e.  A  ->  C  e.  B )   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( C  =  D  <->  x  =  y
 ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  B )
 
Theoremidssen 6922 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 _I  C_  ~~
 
Theoremssdomg 6923 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( B  e.  V  ->  ( A  C_  B  ->  A  ~<_  B ) )
 
Theoremener 6924 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |- 
 ~~  Er  _V
 
Theoremensymb 6925 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~~  B  <->  B 
 ~~  A )
 
Theoremensym 6926 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  ~~  B  ->  B  ~~  A )
 
Theoremensymi 6927 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   =>    |-  B  ~~  A
 
Theoremensymd 6928 Symmetry of equinumerosity. Deduction form of ensym 6926. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  ~~  B )   =>    |-  ( ph  ->  B  ~~  A )
 
Theorementr 6929 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~~  C )  ->  A  ~~  C )
 
Theoremdomtr 6930 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  ( ( A  ~<_  B  /\  B 
 ~<_  C )  ->  A  ~<_  C )
 
Theorementri 6931 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  A  ~~  C
 
Theorementr2i 6932 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  C  ~~  A
 
Theorementr3i 6933 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  A  ~~  C   =>    |-  B  ~~  C
 
Theorementr4i 6934 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  C  ~~  B   =>    |-  A  ~~  C
 
Theoremendomtr 6935 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~<_  C ) 
 ->  A  ~<_  C )
 
Theoremdomentr 6936 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A 
 ~<_  C )
 
Theoremf1imaeng 6937 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C )  ~~  C )
 
Theoremf1imaen2g 6938 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6939 does not need ax-reg 7322.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  ( ( ( F : A -1-1-> B  /\  B  e.  V )  /\  ( C  C_  A  /\  C  e.  V ) )  ->  ( F " C )  ~~  C )
 
Theoremf1imaen 6939 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
 |-  C  e.  _V   =>    |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C ) 
 ~~  C )
 
Theoremen0 6940 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
 |-  ( A  ~~  (/)  <->  A  =  (/) )
 
Theoremensn1 6941 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
 |-  A  e.  _V   =>    |-  { A }  ~~  1o
 
Theoremensn1g 6942 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  { A }  ~~  1o )
 
Theoremenpr1g 6943  { A ,  A } has only one element. (Contributed by FL, 15-Feb-2010.)
 |-  ( A  e.  V  ->  { A ,  A }  ~~  1o )
 
Theoremen1 6944* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
 |-  ( A  ~~  1o  <->  E. x  A  =  { x } )
 
Theoremen1b 6945 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  ( A  ~~  1o  <->  A  =  { U. A }
 )
 
Theoremreuen1 6946* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x  e.  A  ph  <->  { x  e.  A  |  ph }  ~~  1o )
 
Theoremeuen1 6947 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )
 
Theoremeuen1b 6948* Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  1o  <->  E! x  x  e.  A )
 
Theorem2dom 6949* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
 |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
 
Theoremfundmen 6950 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  F  e.  _V   =>    |-  ( Fun  F  ->  dom  F  ~~  F )
 
Theoremfundmeng 6951 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
 |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F  ~~  F )
 
Theoremcnven 6952 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( ( Rel  A  /\  A  e.  V ) 
 ->  A  ~~  `' A )
 
Theoremfndmeng 6953 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
 
Theoremmapsnen 6954 Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  { B } )  ~~  A
 
Theoremmap1 6955 Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)
 |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
 
Theoremen2sn 6956 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A }  ~~  { B } )
 
Theoremsnfi 6957 A singleton is finite. (Contributed by NM, 4-Nov-2002.)
 |- 
 { A }  e.  Fin
 
Theoremfiprc 6958 The class of finite sets is a proper class. (Contributed by Jeffrey Hankins, 3-Oct-2008.)
 |- 
 Fin  e/  _V
 
Theoremunen 6959 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A 
 ~~  B  /\  C  ~~  D )  /\  (
 ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D ) )
 
Theoremdifsnen 6960 All decrements of a set are equinumerous. (Contributed by Stefan O'Rear, 19-Feb-2015.)
 |-  ( ( X  e.  V  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( X  \  { A } )  ~~  ( X  \  { B }
 ) )
 
Theoremdomdifsn 6961 Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  ~<  B  ->  A  ~<_  ( B  \  { C } ) )
 
Theoremxpsnen 6962 A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  { B } )  ~~  A
 
Theoremxpsneng 6963 A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A )
 
Theoremxp1en 6964 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
 
Theoremendisj 6965* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E. x E. y
 ( ( x  ~~  A  /\  y  ~~  B )  /\  ( x  i^i  y )  =  (/) )
 
Theoremundom 6966 Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A  ~<_  B  /\  C  ~<_  D ) 
 /\  ( B  i^i  D )  =  (/) )  ->  ( A  u.  C ) 
 ~<_  ( B  u.  D ) )
 
Theoremxpcomf1o 6967* The canonical bijection from  ( A  X.  B
) to  ( B  X.  A ). (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   =>    |-  F : ( A  X.  B ) -1-1-onto-> ( B  X.  A )
 
Theoremxpcomco 6968* Composition with the bijection of xpcomf1o 6967 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   &    |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )   =>    |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
 
Theoremxpcomen 6969 Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  B )  ~~  ( B  X.  A )
 
Theoremxpcomeng 6970 Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
 
Theoremxpsnen2g 6971 A set is equinumerous to its cross-product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B )
 
Theoremxpassen 6972 Associative law for equinumerosity of cross product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  X.  C )  ~~  ( A  X.  ( B  X.  C ) )
 
Theoremxpdom2 6973 Dominance law for cross product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  C  e.  _V   =>    |-  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) )
 
Theoremxpdom2g 6974 Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  A  ~<_  B ) 
 ->  ( C  X.  A ) 
 ~<_  ( C  X.  B ) )
 
Theoremxpdom1g 6975 Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( C  e.  V  /\  A  ~<_  B ) 
 ->  ( A  X.  C ) 
 ~<_  ( B  X.  C ) )
 
Theoremxpdom3 6976 A set is dominated by its cross product with a non-empty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A 
 ~<_  ( A  X.  B ) )
 
Theoremxpdom1 6977 Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Mar-2006.)
 |-  C  e.  _V   =>    |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
 
Theoremdomunsncan 6978 A singleton cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( -.  A  e.  X  /\  -.  B  e.  Y )  ->  (
 ( { A }  u.  X )  ~<_  ( { B }  u.  Y ) 
 <->  X  ~<_  Y ) )
 
Theoremomxpenlem 6979* Lemma for omxpen 6980. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.)
 |-  F  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )   =>    |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : ( B  X.  A ) -1-1-onto-> ( A  .o  B ) )
 
Theoremomxpen 6980 The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B )  ~~  ( A  X.  B ) )
 
Theoremomf1o 6981* Construct an explicit bijection from  A  .o  B to  B  .o  A. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )   &    |-  G  =  ( x  e.  B ,  y  e.  A  |->  ( ( B  .o  y )  +o  x ) )   =>    |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( G  o.  `' F ) : ( A  .o  B ) -1-1-onto-> ( B  .o  A ) )
 
Theorempw2f1olem 6982* Lemma for pw2f1o 6983. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  (
 ( S  e.  ~P A  /\  G  =  ( z  e.  A  |->  if ( z  e.  S ,  C ,  B ) ) )  <->  ( G  e.  ( { B ,  C }  ^m  A )  /\  S  =  ( `' G " { C }
 ) ) ) )
 
Theorempw2f1o 6983* The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  B  =/=  C )   &    |-  F  =  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) )   =>    |-  ( ph  ->  F : ~P A -1-1-onto-> ( { B ,  C }  ^m  A ) )
 
Theorempw2eng 6984 The power set of a set is equinumerous to set exponentiation with a base of ordinal  2o. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)
 |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
 
Theorempw2en 6985 The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)
 |-  A  e.  _V   =>    |-  ~P A  ~~  ( 2o  ^m  A )
 
Theoremfopwdom 6986 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  e.  _V 
 /\  F : A -onto-> B )  ->  ~P B  ~<_  ~P A )
 
2.4.31  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 6987* Lemma for sbth 6997. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  U. D  C_  ( A  \  (
 g " ( B  \  ( f " U. D ) ) ) )
 
Theoremsbthlem2 6988* Lemma for sbth 6997. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  ( ran  g  C_  A  ->  ( A  \  ( g
 " ( B  \  ( f " U. D ) ) ) )  C_  U. D )
 
Theoremsbthlem3 6989* Lemma for sbth 6997. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  ( ran  g  C_  A  ->  ( g " ( B 
 \  ( f " U. D ) ) )  =  ( A  \  U. D ) )
 
Theoremsbthlem4 6990* Lemma for sbth 6997. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   =>    |-  (
 ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g )  ->  ( `' g " ( A 
 \  U. D ) )  =  ( B  \  ( f " U. D ) ) )
 
Theoremsbthlem5 6991* Lemma for sbth 6997. (Contributed by NM, 22-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( dom  f  =  A  /\  ran  g  C_  A )  ->  dom  H  =  A )
 
Theoremsbthlem6 6992* Lemma for sbth 6997. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( ran  f  C_  B  /\  ( ( dom  g  =  B  /\  ran  g  C_  A )  /\  Fun  `' g
 ) )  ->  ran  H  =  B )
 
Theoremsbthlem7 6993* Lemma for sbth 6997. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( Fun  f  /\  Fun  `' g ) 
 ->  Fun  H )
 
Theoremsbthlem8 6994* Lemma for sbth 6997. (Contributed by NM, 27-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( Fun  `' f  /\  ( ( ( Fun  g  /\  dom  g  =  B )  /\  ran  g  C_  A )  /\  Fun  `' g
 ) )  ->  Fun  `' H )
 
Theoremsbthlem9 6995* Lemma for sbth 6997. (Contributed by NM, 28-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   =>    |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A )  ->  H : A -1-1-onto-> B )
 
Theoremsbthlem10 6996* Lemma for sbth 6997. (Contributed by NM, 28-Mar-1998.)
 |-  A  e.  _V   &    |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f " x ) ) ) 
 C_  ( A  \  x ) ) }   &    |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
 \  U. D ) ) )   &    |-  B  e.  _V   =>    |-  (
 ( A  ~<_  B  /\  B 
 ~<_  A )  ->  A  ~~  B )
 
Theoremsbth 6997 Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 
A is smaller (has lower cardinality) than  B and vice-versa, then  A and  B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 6987 through sbthlem10 6996; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 6996. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. (Contributed by NM, 8-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B 
 ~<_  A )  ->  A  ~~  B )
 
Theoremsbthb 6998 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B 
 ~<_  A )  <->  A  ~~  B )
 
Theoremsbthcl 6999 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  (  ~<_  i^i  `' 
 ~<_  )
 
Theoremdfsdom2 7000 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~<  =  (  ~<_  \  `'  ~<_  )
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