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Theorem List for Metamath Proof Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnfi 7301 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremnndomo 7302 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 7303 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 7304 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 7305 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 7306 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 7307 Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 7306. (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 7308 Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.)
 |-  ( (/)  ~<  A  <->  1o  ~<_  A )
 
Theorem1sdom2 7309 Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.)
 |- 
 1o  ~<  2o
 
Theoremsdom1 7310 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( A  ~<  1o  <->  A  =  (/) )
 
Theoremmodom 7311 Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
 |-  ( E* x ph  <->  { x  |  ph }  ~<_  1o )
 
Theoremmodom2 7312* Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( E* x  x  e.  A  <->  A  ~<_  1o )
 
Theorem1sdom 7313* A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 7181.) (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( A  e.  V  ->  ( 1o  ~<  A  <->  E. x  e.  A  E. y  e.  A  -.  x  =  y
 ) )
 
TheoremfisucdomOLD 7314 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 7315* Lemma for unxpdom 7318. (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 7316* Lemma for unxpdom 7318. (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 7317* Lemma for unxpdom 7318. (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 7318 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 7319 Corollary of unxpdom 7318. (Contributed by NM, 16-Sep-2004.)
 |-  ( ( 1o  ~<  A 
 /\  B  ~<_  A ) 
 ->  ( A  u.  B ) 
 ~<_  ( A  X.  A ) )
 
Theoremsucxpdom 7320 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 7321 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 7322 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 7323 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinf 7324* 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 7325 Lemma for fineqv 7326. (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 7326 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 7327 Equinmerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 7328 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 7329* 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 7330 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A )  ->  B  e.  Fin )
 
Theoremssfi 7331 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 7332 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 7333 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 )
 )
 
Theoreminfi 7334 The interection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( A  e.  Fin  ->  ( A  i^i  B )  e.  Fin )
 
Theoremrabfi 7335* A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( A  e.  Fin  ->  { x  e.  A  |  ph }  e.  Fin )
 
Theoremfinresfin 7336 The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
 |-  ( E  e.  Fin  ->  ( E  |`  B )  e.  Fin )
 
Theoremf1finf1o 7337 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 7338 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremnfielex 7339* If a class is not finite, it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  ( -.  A  e.  Fin 
 ->  E. x  x  e.  A )
 
Theoremen1eqsn 7340 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
 |-  ( ( A  e.  B  /\  B  ~~  1o )  ->  B  =  { A } )
 
Theoremdiffi 7341 If  A is finite,  ( A 
\  B ) is finite. (Contributed by FL, 3-Aug-2009.)
 |-  ( A  e.  Fin  ->  ( A  \  B )  e.  Fin )
 
Theoremdif1enOLD 7342 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 7343 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 7344 Lemma for uses of enp1i 7345. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  T  =  ( { x }  u.  S )   =>    |-  ( x  e.  A  ->  ( ( A  \  { x } )  =  S  ->  A  =  T ) )
 
Theoremenp1i 7345* Proof induction for en2i 7147 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 7346* 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 7347* 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 7348* 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 7349* 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 7350* 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 7351* Variation of findcard2 7350 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 7352* 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 7353* A version of ac6s 8366 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 7354 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 7355* 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 7356* 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 7357* 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 7358 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 7359 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 7360 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 7361* Lemma for unbnn 7365. 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 7362* Lemma for unbnn 7365. 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 7363* Lemma for unbnn 7365. 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 7364* Lemma for unbnn 7365. 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 7365* 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 7615 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 7366* Version of unbnn 7365 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 7367 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 )
 
Theoremnnsdomg 7368 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)
 |-  ( ( om  e.  _V 
 /\  A  e.  om )  ->  A  ~<  om )
 
Theoremisfiniteg 7369 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( om  e.  _V  ->  ( A  e.  Fin  <->  A  ~<  om ) )
 
Theoreminfsdomnn 7370 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B  e.  om )  ->  B  ~<  A )
 
Theoreminfn0 7371 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
 |-  ( om  ~<_  A  ->  A  =/=  (/) )
 
Theoremfin2inf 7372 This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless 
om exists. (Contributed by NM, 13-Nov-2003.)
 |-  ( A  ~<  om  ->  om  e.  _V )
 
Theoremunfilem1 7373* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  F  =  ( x  e.  B  |->  ( A  +o  x ) )   =>    |- 
 ran  F  =  (
 ( A  +o  B )  \  A )
 
Theoremunfilem2 7374* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  F  =  ( x  e.  B  |->  ( A  +o  x ) )   =>    |-  F : B -1-1-onto-> ( ( A  +o  B )  \  A )
 
Theoremunfilem3 7375 Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 16-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  B  ~~  (
 ( A  +o  B )  \  A ) )
 
Theoremunfi 7376 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B )  e.  Fin )
 
Theoremunfir 7377 If a union is finite, the operands are finite. Converse of unfi 7376. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  u.  B )  e.  Fin  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremunfi2 7378 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 7376 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7372). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
 
Theoremdifinf 7379 An infinite set  A minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  -.  ( A  \  B )  e. 
 Fin )
 
Theoremxpfi 7380 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B )  e.  Fin )
 
Theoremdomunfican 7381 A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( ( ( A  e.  Fin  /\  B  ~~  A )  /\  ( ( A  i^i  X )  =  (/)  /\  ( B  i^i  Y )  =  (/) ) )  ->  ( ( A  u.  X )  ~<_  ( B  u.  Y ) 
 <->  X  ~<_  Y ) )
 
Theoreminfcntss 7382* Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
 |-  A  e.  _V   =>    |-  ( om  ~<_  A  ->  E. x ( x  C_  A  /\  x  ~~  om ) )
 
Theoremprfi 7383 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
 |- 
 { A ,  B }  e.  Fin
 
Theoremtpfi 7384 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
 |- 
 { A ,  B ,  C }  e.  Fin
 
Theoremfiint 7385* Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of 
A is in  A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally. (Contributed by NM, 22-Sep-2002.)
 |-  ( A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A  <->  A. x ( ( x 
 C_  A  /\  x  =/= 
 (/)  /\  x  e.  Fin )  ->  |^| x  e.  A ) )
 
Theoremfnfi 7386 A version of fnex 5963 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremfodomfi 7387 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8404 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
 
Theoremfodomfib 7388* Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8406 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)
 |-  ( A  e.  Fin  ->  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B 
 ~<_  A ) ) )
 
Theoremfofinf1o 7389 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
 |-  ( ( F : A -onto-> B  /\  A  ~~  B  /\  B  e.  Fin )  ->  F : A -1-1-onto-> B )
 
Theoremfidomdm 7390 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( F  e.  Fin  ->  dom  F  ~<_  F )
 
Theoremdmfi 7391 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
 |-  ( A  e.  Fin  ->  dom  A  e.  Fin )
 
Theoremcnvfi 7392 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  `' A  e.  Fin )
 
Theoremrnfi 7393 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  ran  A  e.  Fin )
 
Theoremfofi 7394 If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)
 |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
 
Theoremf1fi 7395 If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( B  e.  Fin  /\  F : A -1-1-> B )  ->  A  e.  Fin )
 
Theoremiunfi 7396* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 7397. Note that  B depends on  x, i.e. can be thought of as  B ( x ). (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A  B  e.  Fin )  ->  U_ x  e.  A  B  e.  Fin )
 
Theoremunifi 7397 The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. (Contributed by NM, 22-Aug-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  Fin  /\  A  C_  Fin )  ->  U. A  e.  Fin )
 
Theoremunifi2 7398* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 7397 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7372). (Contributed by NM, 11-Mar-2006.)
 |-  ( ( A  ~<  om 
 /\  A. x  e.  A  x  ~<  om )  ->  U. A  ~<  om )
 
Theoreminfssuni 7399* If an infinite set  A is included in the underlying set of a finite cover  B, then there exists a set of the cover that contains an infinite number of element of  A. (Contributed by FL, 2-Aug-2009.)
 |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin  /\  A  C_  U. B ) 
 ->  E. x  e.  B  -.  ( A  i^i  x )  e.  Fin )
 
Theoremunirnffid 7400 The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  F : T --> Fin )   &    |-  ( ph  ->  T  e.  Fin )   =>    |-  ( ph  ->  U.
 ran  F  e.  Fin )
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