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Theorem List for Metamath Proof Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2pwuninel 6901 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 6902 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
 |-  ( A  e.  V  ->  ~P ~P A  =/=  A )
 
Theoremdisjen 6903 A stronger form of pwuninel 6186. We can use pwuninel 6186, 2pwuninel 6901 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 6904* Existence version of disjen 6903. (Contributed by Mario Carneiro, 7-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. x ( ( A  i^i  x )  =  (/)  /\  x  ~~  B ) )
 
Theoremdomss2 6905 A corollary of disjenex 6904. 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 6906* A corollary of disjenex 6904. 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 6907* Weakening of domssex 6907 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 6908* 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 6909 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 6910 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 6911 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 6912 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 6913* Lemma for xpmapen 6914. (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 6914 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 6915 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 6916 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 6917 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 6918 Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  B  =/=  (/) )  ->  A 
 ~<_  ( A  ^m  B ) )
 
Theorempwen 6919 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 6920* 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 6921 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 6922 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |- 
 Lim  A   =>    |-  ( A  e.  V  ->  A  ~~  suc  A )
 
Theoremlimensuc 6923 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
 |-  ( ( A  e.  V  /\  Lim  A )  ->  A  ~~  suc  A )
 
Theoreminfensuc 6924 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 6925 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 6926 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 6927 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 6928 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 6929 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 6930 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 6925 through phplem4 6928, nneneq 6929, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  -.  A  ~~  B )
 
Theoremphp2 6931 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  B  ~<  A )
 
Theoremphp3 6932 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 6933 Corollary of the Pigeonhole Principle php 6930: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.)
 |-  ( A  e.  om  ->  A  ~<  suc  A )
 
Theoremphp5 6934 Corollary of the Pigeonhole Principle php 6930: 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 6935 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 6936 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 6937 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 6938 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremnndomo 6939 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 6940 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 6941 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 6942 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 6943 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 6944 Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 6943. (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 6945 Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.)
 |-  ( (/)  ~<  A  <->  1o  ~<_  A )
 
Theorem1sdom2 6946 Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.)
 |- 
 1o  ~<  2o
 
Theoremsdom1 6947 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A  ~<  1o  <->  A  =  (/) )
 
Theoremmodom 6948 Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E* x ph  <->  { x  |  ph }  ~<_  1o )
 
Theoremmodom2 6949* Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E* x  x  e.  A  <->  A  ~<_  1o )
 
Theorem1sdom 6950* A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 6818.) (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( A  e.  V  ->  ( 1o  ~<  A  <->  E. x  e.  A  E. y  e.  A  -.  x  =  y
 ) )
 
TheoremfisucdomOLD 6951 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 6952* Lemma for unxpdom 6955. (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 >. ) )
 
Theoremunxpdomlem2 6953* Lemma for unxpdom 6955. (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 >. )   &    |-  ( ph  ->  w  e.  ( a  u.  b ) )   &    |-  ( ph  ->  -.  m  =  n )   &    |-  ( ph  ->  -.  s  =  t )   =>    |-  ( ( ph  /\  (
 z  e.  a  /\  -.  w  e.  a ) )  ->  -.  ( F `  z )  =  ( F `  w ) )
 
Theoremunxpdomlem3 6954* Lemma for unxpdom 6955. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  F  =  ( x  e.  ( a  u.  b )  |->  G )   &    |-  G  =  if ( x  e.  a ,  <. x ,  if ( x  =  m ,  t ,  s ) >. ,  <. if ( x  =  t ,  n ,  m ) ,  x >. )   =>    |-  ( ( 1o  ~<  a 
 /\  1o  ~<  b ) 
 ->  ( a  u.  b
 )  ~<_  ( a  X.  b ) )
 
Theoremunxpdom 6955 Cross product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( 1o  ~<  A 
 /\  1o  ~<  B ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  B ) )
 
Theoremunxpdom2 6956 Corollary of unxpdom 6955. (Contributed by NM, 16-Sep-2004.)
 |-  ( ( 1o  ~<  A 
 /\  B  ~<_  A ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  A ) )
 
Theoremsucxpdom 6957 Cross product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
 |-  ( 1o  ~<  A  ->  suc 
 A  ~<_  ( A  X.  A ) )
 
Theorempssinf 6958 A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |-  ( ( A  C.  B  /\  A  ~~  B )  ->  -.  B  e.  Fin )
 
Theoremfisseneq 6959 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
 |-  ( ( B  e.  Fin  /\  A  C_  B  /\  A  ~~  B )  ->  A  =  B )
 
Theoremominf 6960 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinf 6961* Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.)
 |-  ( -.  A  e.  Fin 
 ->  A. n  e.  om  E. x ( x  C_  A  /\  x  ~~  n ) )
 
Theoremfineqvlem 6962 Lemma for fineqv 6963. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  om  ~<_  ~P ~P A )
 
Theoremfineqv 6963 If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.)
 |-  ( -.  om  e.  _V  <->  Fin 
 =  _V )
 
Theoremenfi 6964 Equinmerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 6965 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( B  e.  Fin  /\  A  ~~  B ) 
 ->  A  e.  Fin )
 
Theorempssnn 6966* A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  om 
 /\  B  C.  A )  ->  E. x  e.  A  B  ~~  x )
 
Theoremssnnfi 6967 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremssfi 6968 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  Fin  /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremdomfi 6969 A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  ~<_  A )  ->  B  e.  Fin )
 
Theoremxpfir 6970 The components of a non-empty finite cross product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( ( A  X.  B )  e. 
 Fin  /\  ( A  X.  B )  =/=  (/) )  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremf1finf1o 6971 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)
 |-  ( ( A  ~~  B  /\  B  e.  Fin )  ->  ( F : A -1-1-> B  <->  F : A -1-1-onto-> B ) )
 
Theorem0fin 6972 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremen1eqsn 6973 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
 |-  ( ( A  e.  B  /\  B  ~~  1o )  ->  B  =  { A } )
 
Theoremdiffi 6974 If  A is finite,  ( A 
\  B ) is finite. (Contributed by FL, 3-Aug-2009.)
 |-  ( A  e.  Fin  ->  ( A  \  B )  e.  Fin )
 
Theoremdif1enOLD 6975 If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( M  e.  om  /\  A  ~~ 
 suc  M  /\  X  e.  A )  ->  ( A 
 \  { X }
 )  ~~  M )
 
Theoremdif1en 6976 If a set  A is equinumerous to the successor of a natural number  M, then  A with an element removed is equinumerous to  M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
 |-  ( ( M  e.  om 
 /\  A  ~~  suc  M 
 /\  X  e.  A )  ->  ( A  \  { X } )  ~~  M )
 
Theoremenp1ilem 6977 Lemma for uses of enp1i 6978. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  T  =  ( { x }  u.  S )   =>    |-  ( x  e.  A  ->  ( ( A  \  { x } )  =  S  ->  A  =  T ) )
 
Theoremenp1i 6978* Proof induction for en2i 6785 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  M  e.  om   &    |-  N  =  suc  M   &    |-  ( ( A 
 \  { x }
 )  ~~  M  ->  ph )   &    |-  ( x  e.  A  ->  ( ph  ->  ps ) )   =>    |-  ( A  ~~  N  ->  E. x ps )
 
Theoremen2 6979* A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  2o  ->  E. x E. y  A  =  { x ,  y } )
 
Theoremen3 6980* A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  3o  ->  E. x E. y E. z  A  =  { x ,  y ,  z } )
 
Theoremen4 6981* A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( A  ~~  4o  ->  E. x E. y E. z E. w  A  =  ( { x ,  y }  u.  { z ,  w } ) )
 
Theoremfindcard 6982* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  ( y  \  { z } )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  (
 A. z  e.  y  ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard2 6983* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  Fin  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  Fin  ->  ta )
 
Theoremfindcard2s 6984* Variation of findcard2 6983 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( y  e.  Fin  /\ 
 -.  z  e.  y
 )  ->  ( ch  ->  th ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremfindcard3 6985* Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.)
 |-  ( x  =  y 
 ->  ( ph  <->  ch ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  (
 y  e.  Fin  ->  (
 A. x ( x 
 C.  y  ->  ph )  ->  ch ) )   =>    |-  ( A  e.  Fin 
 ->  ta )
 
Theoremac6sfi 6986* A version of ac6s 7995 for finite sets. (Contributed by Jeffrey Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  Fin  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremfrfi 6987 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Po  A  /\  A  e.  Fin )  ->  R  Fr  A )
 
Theoremfimax2g 6988* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremfimaxg 6989* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  y R x ) )
 
Theoremfisupg 6990* Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  A  y R z ) ) )
 
Theoremwofi 6991 A total order on a finite set is a well order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
 
Theoremordunifi 6992 The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  U. A  e.  A )
 
Theoremnnunifi 6993 The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ( S  C_  om 
 /\  S  e.  Fin )  ->  U. S  e.  om )
 
Theoremunblem1 6994* Lemma for unbnn 6998. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)
 |-  ( ( ( B 
 C_  om  /\  A. x  e.  om  E. y  e.  B  x  e.  y
 )  /\  A  e.  B )  ->  |^| ( B  \  suc  A )  e.  B )
 
Theoremunblem2 6995* Lemma for unbnn 6998. The value of the function  F belongs to the unbounded set of natural numbers  A. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
 z  e.  om  ->  ( F `  z )  e.  A ) )
 
Theoremunblem3 6996* Lemma for unbnn 6998. The value of the function  F is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
 z  e.  om  ->  ( F `  z )  e.  ( F `  suc  z ) ) )
 
Theoremunblem4 6997* Lemma for unbnn 6998. The function  F maps the set of natural numbers one-to-one to the set of unbounded natural numbers  A. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  F : om -1-1-> A )
 
Theoremunbnn 6998* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 7243 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
 |-  ( ( om  e.  _V 
 /\  A  C_  om  /\  A. x  e.  om  E. y  e.  A  x  e.  y )  ->  A  ~~ 
 om )
 
Theoremunbnn2 6999* Version of unbnn 6998 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)
 |-  ( ( om  e.  _V 
 /\  A  C_  om  /\  A. x  e.  om  E. y  e.  A  x  C_  y )  ->  A  ~~ 
 om )
 
Theoremisfinite2 7000 Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  ~<  om  ->  A  e.  Fin )
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