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Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorementr2i 6801 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  B  ~~  C   =>    |-  C  ~~  A
 
Theorementr3i 6802 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  A  ~~  C   =>    |-  B  ~~  C
 
Theorementr4i 6803 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
 |-  A  ~~  B   &    |-  C  ~~  B   =>    |-  A  ~~  C
 
Theoremendomtr 6804 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~~  B  /\  B  ~<_  C ) 
 ->  A  ~<_  C )
 
Theoremdomentr 6805 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B  ~~  C )  ->  A 
 ~<_  C )
 
Theoremf1imaeng 6806 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 6807 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6808 does not need ax-reg 7190.) (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 6808 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 6809 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
 |-  ( A  ~~  (/)  <->  A  =  (/) )
 
Theoremensn1 6810 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
 |-  A  e.  _V   =>    |-  { A }  ~~  1o
 
Theoremensn1g 6811 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
 |-  ( A  e.  V  ->  { A }  ~~  1o )
 
Theoremenpr1g 6812  { A ,  A } has only one element. (Contributed by FL, 15-Feb-2010.)
 |-  ( A  e.  V  ->  { A ,  A }  ~~  1o )
 
Theoremen1 6813* 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 6814 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 6815* 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 6816 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )
 
Theoremeuen1b 6817* Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  1o  <->  E! x  x  e.  A )
 
Theorem2dom 6818* 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 6819 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 6820 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 6821 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( ( Rel  A  /\  A  e.  V ) 
 ->  A  ~~  `' A )
 
Theoremfndmeng 6822 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
 
Theoremmapsnen 6823 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 6824 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 6825 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A }  ~~  { B } )
 
Theoremsnfi 6826 A singleton is finite. (Contributed by NM, 4-Nov-2002.)
 |- 
 { A }  e.  Fin
 
Theoremfiprc 6827 The class of finite sets is a proper class. (Contributed by Jeffrey Hankins, 3-Oct-2008.)
 |- 
 Fin  e/  _V
 
Theoremunen 6828 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 6829 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 6830 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 6831 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 6832 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 6833 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 6834* 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 6835 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 6836* 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 6837* Composition with the bijection of xpcomf1o 6836 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 6838 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 6839 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 6840 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 6841 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 6842 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 6843 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 6844 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 6845 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 6846 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 6847 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 6848* Lemma for omxpen 6849. (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 6849 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 6850* 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 6851* Lemma for pw2f1o 6852. (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 6852* 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 6853 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 6854 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 6855 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 )
 
2.4.29  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 6856* Lemma for sbth 6866. (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 6857* Lemma for sbth 6866. (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 6858* Lemma for sbth 6866. (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 6859* Lemma for sbth 6866. (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 6860* Lemma for sbth 6866. (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 6861* Lemma for sbth 6866. (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 6862* Lemma for sbth 6866. (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 6863* Lemma for sbth 6866. (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 6864* Lemma for sbth 6866. (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 6865* Lemma for sbth 6866. (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 6866 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 6856 through sbthlem10 6865; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 6865. 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 6867 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B 
 ~<_  A )  <->  A  ~~  B )
 
Theoremsbthcl 6868 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  (  ~<_  i^i  `' 
 ~<_  )
 
Theoremdfsdom2 6869 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~<  =  (  ~<_  \  `'  ~<_  )
 
Theorembrsdom2 6870 Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ~<  B  <->  ( A  ~<_  B  /\  -.  B  ~<_  A ) )
 
Theoremsdomnsym 6871 Strict dominance is not symmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
 |-  ( A  ~<  B  ->  -.  B  ~<  A )
 
Theoremdomnsym 6872 Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<_  B  ->  -.  B  ~<  A )
 
Theorem0domg 6873 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  -> 
 (/)  ~<_  A )
 
Theoremdom0 6874 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
 |-  ( A  ~<_  (/)  <->  A  =  (/) )
 
Theorem0sdomg 6875 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.)
 |-  ( A  e.  V  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
 
Theorem0dom 6876 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  (/)  ~<_  A
 
Theorem0sdom 6877 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( (/)  ~<  A  <->  A  =/=  (/) )
 
Theoremsdom0 6878 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.)
 |- 
 -.  A  ~<  (/)
 
Theoremsdomdomtr 6879 Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~<  B 
 /\  B  ~<_  C ) 
 ->  A  ~<  C )
 
Theoremsdomentr 6880 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
 |-  ( ( A  ~<  B 
 /\  B  ~~  C )  ->  A  ~<  C )
 
Theoremdomsdomtr 6881 Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~<_  B  /\  B  ~<  C )  ->  A  ~<  C )
 
Theoremensdomtr 6882 Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~~  B  /\  B  ~<  C ) 
 ->  A  ~<  C )
 
Theoremsdomirr 6883 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
 |- 
 -.  A  ~<  A
 
Theoremsdomtr 6884 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
 |-  ( ( A  ~<  B 
 /\  B  ~<  C ) 
 ->  A  ~<  C )
 
Theoremsdomn2lp 6885 Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
 |- 
 -.  ( A  ~<  B 
 /\  B  ~<  A )
 
Theoremenen1 6886 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( A  ~~  C  <->  B 
 ~~  C ) )
 
Theoremenen2 6887 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( C  ~~  A  <->  C 
 ~~  B ) )
 
Theoremdomen1 6888 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<_  C  <->  B  ~<_  C )
 )
 
Theoremdomen2 6889 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B )
 )
 
Theoremsdomen1 6890 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )
 
Theoremsdomen2 6891 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<  A  <->  C  ~<  B ) )
 
Theoremdomtriord 6892 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
 
Theoremsdomel 6893 Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<  B 
 ->  A  e.  B ) )
 
Theoremsdomdif 6894 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
 |-  ( A  ~<  B  ->  ( B  \  A )  =/=  (/) )
 
Theoremonsdominel 6895 An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.)
 |-  ( ( A  e.  On  /\  B  e.  On  /\  ( A  i^i  C )  ~<  ( B  i^i  C ) )  ->  A  e.  B )
 
Theoremdomunsn 6896 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~<  B  ->  ( A  u.  { C } )  ~<_  B )
 
Theoremfodomr 6897* There exists a mapping from a set onto any (non-empty) set that it dominates. (Contributed by NM, 23-Mar-2006.)
 |-  ( ( (/)  ~<  B  /\  B 
 ~<_  A )  ->  E. f  f : A -onto-> B )
 
Theorempwdom 6898 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
 
Theoremcanth2 6899 Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 6178. (Contributed by NM, 7-Aug-1994.)
 |-  A  e.  _V   =>    |-  A  ~<  ~P A
 
Theoremcanth2g 6900 Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.)
 |-  ( A  e.  V  ->  A  ~<  ~P A )
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