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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempw2eng 6901 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 6902 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 6903 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 6904* Lemma for sbth 6914. (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 6905* Lemma for sbth 6914. (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 6906* Lemma for sbth 6914. (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 6907* Lemma for sbth 6914. (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 6908* Lemma for sbth 6914. (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 6909* Lemma for sbth 6914. (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 6910* Lemma for sbth 6914. (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 6911* Lemma for sbth 6914. (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 6912* Lemma for sbth 6914. (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 6913* Lemma for sbth 6914. (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 6914 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 6904 through sbthlem10 6913; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 6913. 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 6915 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)
 |-  ( ( A  ~<_  B  /\  B 
 ~<_  A )  <->  A  ~~  B )
 
Theoremsbthcl 6916 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
 |- 
 ~~  =  (  ~<_  i^i  `' 
 ~<_  )
 
Theoremdfsdom2 6917 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
 |- 
 ~<  =  (  ~<_  \  `'  ~<_  )
 
Theorembrsdom2 6918 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 6919 Strict dominance is not symmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
 |-  ( A  ~<  B  ->  -.  B  ~<  A )
 
Theoremdomnsym 6920 Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)
 |-  ( A  ~<_  B  ->  -.  B  ~<  A )
 
Theorem0domg 6921 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  -> 
 (/)  ~<_  A )
 
Theoremdom0 6922 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
 |-  ( A  ~<_  (/)  <->  A  =  (/) )
 
Theorem0sdomg 6923 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.)
 |-  ( A  e.  V  ->  ( (/)  ~<  A  <->  A  =/=  (/) ) )
 
Theorem0dom 6924 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  (/)  ~<_  A
 
Theorem0sdom 6925 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
 |-  A  e.  _V   =>    |-  ( (/)  ~<  A  <->  A  =/=  (/) )
 
Theoremsdom0 6926 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.)
 |- 
 -.  A  ~<  (/)
 
Theoremsdomdomtr 6927 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 6928 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 6929 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 6930 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 6931 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
 |- 
 -.  A  ~<  A
 
Theoremsdomtr 6932 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
 |-  ( ( A  ~<  B 
 /\  B  ~<  C ) 
 ->  A  ~<  C )
 
Theoremsdomn2lp 6933 Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
 |- 
 -.  ( A  ~<  B 
 /\  B  ~<  A )
 
Theoremenen1 6934 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( A  ~~  C  <->  B 
 ~~  C ) )
 
Theoremenen2 6935 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( C  ~~  A  <->  C 
 ~~  B ) )
 
Theoremdomen1 6936 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<_  C  <->  B  ~<_  C )
 )
 
Theoremdomen2 6937 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B )
 )
 
Theoremsdomen1 6938 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )
 
Theoremsdomen2 6939 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<  A  <->  C  ~<  B ) )
 
Theoremdomtriord 6940 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 6941 Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ~<  B 
 ->  A  e.  B ) )
 
Theoremsdomdif 6942 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
 |-  ( A  ~<  B  ->  ( B  \  A )  =/=  (/) )
 
Theoremonsdominel 6943 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 6944 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  ( A  ~<  B  ->  ( A  u.  { C } )  ~<_  B )
 
Theoremfodomr 6945* 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 6946 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~<_  B  ->  ~P A  ~<_  ~P B )
 
Theoremcanth2 6947 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 6225. (Contributed by NM, 7-Aug-1994.)
 |-  A  e.  _V   =>    |-  A  ~<  ~P A
 
Theoremcanth2g 6948 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 )
 
Theorem2pwuninel 6949 The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.)
 |- 
 -.  ~P ~P U. A  e.  A
 
Theorem2pwne 6950 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
 |-  ( A  e.  V  ->  ~P ~P A  =/=  A )
 
Theoremdisjen 6951 A stronger form of pwuninel 6233. We can use pwuninel 6233, 2pwuninel 6949 to create one or two sets disjoint from a given set  A, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set  B we can construct a set  x that is equinumerous to it and disjoint from  A. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  i^i  ( B  X.  { ~P U. ran  A } ) )  =  (/)  /\  ( B  X.  { ~P U. ran  A } )  ~~  B ) )
 
Theoremdisjenex 6952* Existence version of disjen 6951. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x ( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) )
 
Theoremdomss2 6953 A corollary of disjenex 6952. If  F is an injection from  A to  B then  G is a right inverse of  F from  B to a superset of  A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  G  =  `' ( F  u.  ( 1st  |`  ( ( B  \  ran  F )  X.  { ~P U. ran  A } ) ) )   =>    |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W )  ->  ( G : B
 -1-1-onto-> ran  G  /\  A  C_  ran 
 G  /\  ( G  o.  F )  =  (  _I  |`  A )
 ) )
 
Theoremdomssex2 6954* A corollary of disjenex 6952. If  F is an injection from  A to  B then there is a right inverse  g of  F from  B to a superset of  A. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W )  ->  E. g ( g : B -1-1-> _V  /\  ( g  o.  F )  =  (  _I  |`  A ) ) )
 
Theoremdomssex 6955* Weakening of domssex 6955 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( A  ~<_  B  ->  E. x ( A  C_  x  /\  B  ~~  x ) )
 
2.4.30  Equinumerosity (cont.)
 
Theoremxpf1o 6956* Construct a bijection on a cross product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  ( ph  ->  ( x  e.  A  |->  X ) : A -1-1-onto-> B )   &    |-  ( ph  ->  ( y  e.  C  |->  Y ) : C -1-1-onto-> D )   =>    |-  ( ph  ->  ( x  e.  A ,  y  e.  C  |->  <. X ,  Y >. ) : ( A  X.  C ) -1-1-onto-> ( B  X.  D ) )
 
Theoremxpen 6957 Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
 
Theoremmapen 6958 Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  ^m  C )  ~~  ( B 
 ^m  D ) )
 
Theoremmapdom1 6959 Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
 |-  ( A  ~<_  B  ->  ( A  ^m  C )  ~<_  ( B  ^m  C ) )
 
Theoremmapxpen 6960 Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X ) 
 ->  ( ( A  ^m  B )  ^m  C ) 
 ~~  ( A  ^m  ( B  X.  C ) ) )
 
Theoremxpmapenlem 6961* Lemma for xpmapen 6962. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  =  ( z  e.  C  |->  ( 1st `  ( x `  z ) ) )   &    |-  R  =  ( z  e.  C  |->  ( 2nd `  ( x `  z ) ) )   &    |-  S  =  ( z  e.  C  |->  <.
 ( ( 1st `  y
 ) `  z ) ,  ( ( 2nd `  y
 ) `  z ) >. )   =>    |-  ( ( A  X.  B )  ^m  C ) 
 ~~  ( ( A 
 ^m  C )  X.  ( B  ^m  C ) )
 
Theoremxpmapen 6962 Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  ^m  C )  ~~  ( ( A  ^m  C )  X.  ( B  ^m  C ) )
 
Theoremmapunen 6963 Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  /\  ( A  i^i  B )  =  (/) )  ->  ( C 
 ^m  ( A  u.  B ) )  ~~  ( ( C  ^m  A )  X.  ( C  ^m  B ) ) )
 
Theoremmap2xp 6964 A cardinal power with exponent 2 is equivalent to a cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
 |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A ) )
 
Theoremmapdom2 6965 Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  ~<_  B  /\  -.  ( A  =  (/)  /\  C  =  (/) ) ) 
 ->  ( C  ^m  A ) 
 ~<_  ( C  ^m  B ) )
 
Theoremmapdom3 6966 Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A 
 ~<_  ( A  ^m  B ) )
 
Theorempwen 6967 If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)
 |-  ( A  ~~  B  ->  ~P A  ~~  ~P B )
 
Theoremssenen 6968* Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( A  ~~  B  ->  { x  |  ( x  C_  A  /\  x  ~~  C ) }  ~~  { x  |  ( x  C_  B  /\  x  ~~  C ) }
 )
 
Theoremlimenpsi 6969 A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |- 
 Lim  A   =>    |-  ( A  e.  V  ->  A  ~~  ( A 
 \  { (/) } )
 )
 
Theoremlimensuci 6970 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |- 
 Lim  A   =>    |-  ( A  e.  V  ->  A  ~~  suc  A )
 
Theoremlimensuc 6971 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |-  ( ( A  e.  V  /\  Lim  A )  ->  A  ~~  suc  A )
 
Theoreminfensuc 6972 Any infinite ordinal is equinumerous to its successor. Exercise 7 of [TakeutiZaring] p. 88. Proved without the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 13-Jan-2013.)
 |-  ( ( A  e.  On  /\  om  C_  A )  ->  A  ~~  suc  A )
 
2.4.31  Pigeonhole Principle
 
Theoremphplem1 6973 Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  ( { A }  u.  ( A  \  { B } ) )  =  ( suc  A  \  { B } )
 )
 
Theoremphplem2 6974 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem3 6975 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem4 6976 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  ~~ 
 suc  B  ->  A  ~~  B ) )
 
Theoremnneneq 6977 Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~~  B 
 <->  A  =  B ) )
 
Theoremphp 6978 Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6973 through phplem4 6976, nneneq 6977, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  -.  A  ~~  B )
 
Theoremphp2 6979 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  B  ~<  A )
 
Theoremphp3 6980 Corollary of Pigeonhole Principle. If  A is finite and 
B is a proper subset of  A, the  B is strictly less numerous than  A. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.)
 |-  ( ( A  e.  Fin  /\  B  C.  A ) 
 ->  B  ~<  A )
 
Theoremphp4 6981 Corollary of the Pigeonhole Principle php 6978: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  om  ->  A  ~<  suc  A )
 
Theoremphp5 6982 Corollary of the Pigeonhole Principle php 6978: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  om  ->  -.  A  ~~  suc  A )
 
2.4.32  Finite sets
 
Theoremonomeneq 6983 An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B 
 <->  A  =  B ) )
 
Theoremonfin 6984 An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  On  ->  ( A  e.  Fin  <->  A  e.  om ) )
 
Theoremonfin2 6985 A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
 |- 
 om  =  ( On 
 i^i  Fin )
 
Theoremnnfi 6986 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremnndomo 6987 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<_  B  <->  A  C_  B ) )
 
Theoremnnsdomo 6988 Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<  B  <->  A  C.  B ) )
 
TheoremomsucdomOLD 6989 Strict dominance of natural numbers is the same as dominance over the successor of the smaller. (Contributed by NM, 25-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
 
Theoremsucdom2 6990 Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( A  ~<  B  ->  suc 
 A  ~<_  B )
 
Theoremsucdom 6991 Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( A  e.  om  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
 
TheoremsucdomiOLD 6992 Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 6991. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  om 
 /\  B  e.  C )  ->  ( suc  A  ~<_  B  ->  A  ~<  B ) )
 
Theorem0sdom1dom 6993 Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.)
 |-  ( (/)  ~<  A  <->  1o  ~<_  A )
 
Theorem1sdom2 6994 Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.)
 |- 
 1o  ~<  2o
 
Theoremsdom1 6995 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A  ~<  1o  <->  A  =  (/) )
 
Theoremmodom 6996 Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E* x ph  <->  { x  |  ph }  ~<_  1o )
 
Theoremmodom2 6997* Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E* x  x  e.  A  <->  A  ~<_  1o )
 
Theorem1sdom 6998* A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 6866.) (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( A  e.  V  ->  ( 1o  ~<  A  <->  E. x  e.  A  E. y  e.  A  -.  x  =  y
 ) )
 
TheoremfisucdomOLD 6999 Strict dominance of a finite set over a natural number is the same as dominance over its successor. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( A  e.  om 
 /\  B  e.  Fin )  ->  ( A  ~<  B  <->  suc  A  ~<_  B ) )
 
Theoremunxpdomlem1 7000* Lemma for unxpdom 7003. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   =>    |-  ( z  e.  (
 a  u.  b ) 
 ->  ( F `  z
 )  =  if (
 z  e.  a , 
 <. z ,  if (
 z  =  m ,  t ,  s ) >. ,  <. if ( z  =  t ,  n ,  m ) ,  z >. ) )
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