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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpdom2 6501 Dominance law for Cartesian 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 6502 Dominance law for Cartesian 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 6503 Dominance law for Cartesian 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 ) )
 
Theoremxpdom3m 6504* A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  E. x  x  e.  B )  ->  A  ~<_  ( A  X.  B ) )
 
Theoremxpdom1 6505 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
 |-  C  e.  _V   =>    |-  ( A  ~<_  B  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
 
Theoremfopwdom 6506 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 )
 
Theorem0domg 6507 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  -> 
 (/)  ~<_  A )
 
Theoremdom0 6508 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
 |-  ( A  ~<_  (/)  <->  A  =  (/) )
 
Theorem0dom 6509 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  (/)  ~<_  A
 
Theoremenen1 6510 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( A  ~~  C  <->  B 
 ~~  C ) )
 
Theoremenen2 6511 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( C  ~~  A  <->  C 
 ~~  B ) )
 
Theoremdomen1 6512 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<_  C  <->  B  ~<_  C )
 )
 
Theoremdomen2 6513 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B )
 )
 
2.6.27  Equinumerosity (cont.)
 
Theoremxpf1o 6514* Construct a bijection on a Cartesian 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 6515 Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
 
Theoremmapen 6516 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 ) )
 
Theoremmapdom1g 6517 Order-preserving property of set exponentiation. (Contributed by Jim Kingdon, 15-Jul-2022.)
 |-  ( ( A  ~<_  B  /\  C  e.  V )  ->  ( A  ^m  C ) 
 ~<_  ( B  ^m  C ) )
 
Theoremmapxpen 6518 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 6519* Lemma for xpmapen 6520. (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 6520 Equinumerosity law for set exponentiation of a Cartesian 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 ) )
 
Theoremssenen 6521* 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 ) }
 )
 
2.6.28  Pigeonhole Principle
 
Theoremphplem1 6522 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 6523 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 6524 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6526. (Contributed by NM, 26-May-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem4 6525 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 ) )
 
Theoremphplem3g 6526 A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6524 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremnneneq 6527 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 ) )
 
Theoremphp5 6528 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 )
 
Theoremsnnen2og 6529 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a proper class, see snnen2oprc 6530. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  V  ->  -.  { A }  ~~  2o )
 
Theoremsnnen2oprc 6530 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a set, see snnen2og 6529. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( -.  A  e.  _V 
 ->  -.  { A }  ~~  2o )
 
Theorem1nen2 6531 One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
 |- 
 -.  1o  ~~  2o
 
Theoremphplem4dom 6532 Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  ~<_  suc  B  ->  A  ~<_  B ) )
 
Theoremphp5dom 6533 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
 
Theoremnndomo 6534 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 ) )
 
Theoremphpm 6535* Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols  E. x x  e.  ( A  \  B
) (which is stronger than not being equal in the absence of excluded middle). 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 6522 through phplem4 6525, nneneq 6527, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
 |-  ( ( A  e.  om 
 /\  B  C_  A  /\  E. x  x  e.  ( A  \  B ) )  ->  -.  A  ~~  B )
 
Theoremphpelm 6536 Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  A )  ->  -.  A  ~~  B )
 
Theoremphplem4on 6537 Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
 |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( suc  A  ~~ 
 suc  B  ->  A  ~~  B ) )
 
2.6.29  Finite sets
 
Theoremfict 6538 A finite set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.)
 |-  ( A  e.  Fin  ->  A 
 ~<_  om )
 
Theoremfidceq 6539 Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that  { B ,  C } is finite would require showing it is equinumerous to  1o or to  2o but to show that you'd need to know  B  =  C or  -.  B  =  C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A  /\  C  e.  A )  -> DECID  B  =  C )
 
Theoremfidifsnen 6540 All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
 |-  ( ( X  e.  Fin  /\  A  e.  X  /\  B  e.  X )  ->  ( X  \  { A } )  ~~  ( X  \  { B }
 ) )
 
Theoremfidifsnid 6541 If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3568 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( ( A  \  { B } )  u. 
 { B } )  =  A )
 
Theoremnnfi 6542 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremenfi 6543 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 6544 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( B  e.  Fin  /\  A  ~~  B ) 
 ->  A  e.  Fin )
 
Theoremssfilem 6545* Lemma for ssfiexmid 6546. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |- 
 { z  e.  { (/)
 }  |  ph }  e.  Fin   =>    |-  ( ph  \/  -.  ph )
 
Theoremssfiexmid 6546* If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
 |- 
 A. x A. y
 ( ( x  e. 
 Fin  /\  y  C_  x )  ->  y  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoreminfiexmid 6547* If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( x  e.  Fin  ->  ( x  i^i  y )  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremdomfiexmid 6548* If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |-  ( ( x  e. 
 Fin  /\  y  ~<_  x ) 
 ->  y  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremdif1en 6549 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 )
 
Theoremdif1enen 6550 Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A 
 ~~  B )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  D  e.  B )   =>    |-  ( ph  ->  ( A  \  { C }
 )  ~~  ( B  \  { D } )
 )
 
Theoremfiunsnnn 6551 Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
 |-  ( ( ( A  e.  Fin  /\  B  e.  ( _V  \  A ) )  /\  ( N  e.  om  /\  A  ~~  N ) )  ->  ( A  u.  { B } )  ~~  suc  N )
 
Theoremphp5fin 6552 A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  ( _V  \  A ) )  ->  -.  A  ~~  ( A  u.  { B }
 ) )
 
Theoremfisbth 6553 Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
 |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B )
 
Theorem0fin 6554 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremfin0 6555* A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
 |-  ( A  e.  Fin  ->  ( A  =/=  (/)  <->  E. x  x  e.  A ) )
 
Theoremfin0or 6556* A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
 |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  E. x  x  e.  A ) )
 
Theoremdiffitest 6557* If subtracting any set from a finite set gives a finite set, any proposition of the form  -.  ph is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove  A  e.  Fin  ->  ( A  \  B
)  e.  Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
 |- 
 A. a  e.  Fin  A. b ( a  \  b )  e.  Fin   =>    |-  ( -.  ph  \/  -.  -.  ph )
 
Theoremfindcard 6558* 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 6559* 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 6560* Variation of findcard2 6559 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 )
 
Theoremfindcard2d 6561* Deduction version of findcard2 6559. If you also need  y  e.  Fin (which doesn't come for free due to ssfiexmid 6546), use findcard2sd 6562 instead. (Contributed by SO, 16-Jul-2018.)
 |-  ( x  =  (/)  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  ( ps  <->  th ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  ( y  C_  A  /\  z  e.  ( A  \  y ) ) )  ->  ( th  ->  ta ) )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  et )
 
Theoremfindcard2sd 6562* Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
 |-  ( x  =  (/)  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  ( ps  <->  th ) )   &    |-  ( x  =  ( y  u.  { z } )  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  ( ph  ->  ch )   &    |-  ( ( (
 ph  /\  y  e.  Fin )  /\  ( y 
 C_  A  /\  z  e.  ( A  \  y
 ) ) )  ->  ( th  ->  ta )
 )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  et )
 
Theoremdiffisn 6563 Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( A  \  { B } )  e.  Fin )
 
Theoremdiffifi 6564 Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( A  \  B )  e.  Fin )
 
Theoreminfnfi 6565 An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  -.  A  e.  Fin )
 
Theoremominf 6566 The set of natural numbers is not finite. Although we supply this theorem because we can, the more natural way to express " om is infinite" is  om  ~<_  om which is an instance of domrefg 6438. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinfinf 6567* An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)
 |-  ( om  ~<_  A  ->  A. n  e.  om  E. x ( x  C_  A  /\  x  ~~  n ) )
 
Theoremac6sfi 6568* Existence of a choice function for finite sets. (Contributed by Jeff 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 ) )
 
Theoremtridc 6569* A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  -> DECID  B R C )
 
Theoremfimax2gtrilemstep 6570* Lemma for fimax2gtri 6571. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  U  C_  A )   &    |-  ( ph  ->  Z  e.  A )   &    |-  ( ph  ->  V  e.  A )   &    |-  ( ph  ->  -.  V  e.  U )   &    |-  ( ph  ->  A. y  e.  U  -.  Z R y )   =>    |-  ( ph  ->  E. x  e.  A  A. y  e.  ( U  u.  { V } )  -.  x R y )
 
Theoremfimax2gtri 6571* A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.)
 |-  ( ph  ->  R  Po  A )   &    |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/= 
 (/) )   =>    |-  ( ph  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremfinexdc 6572* Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  -> DECID  E. x  e.  A  ph )
 
Theoremdfrex2fin 6573* Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A DECID  ph )  ->  ( E. x  e.  A  ph  <->  -.  A. x  e.  A  -.  ph )
 )
 
Theoreminfm 6574* An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
 |-  ( om  ~<_  A  ->  E. x  x  e.  A )
 
Theoreminfn0 6575 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
 |-  ( om  ~<_  A  ->  A  =/=  (/) )
 
Theoreminffiexmid 6576* If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.)
 |-  ( x  e.  Fin  \/ 
 om  ~<_  x )   =>    |-  ( ph  \/  -.  ph )
 
Theoremen2eqpr 6577 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C ) 
 ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
 
Theoremexmidpw 6578 Excluded middle is equivalent to the power set of  1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
 |-  (EXMID  <->  ~P 1o  ~~  2o )
 
Theoremfientri3 6579 Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ~<_  B  \/  B 
 ~<_  A ) )
 
Theoremnnwetri 6580* A natural number is well-ordered by 
_E. More specifically, this order both satisfies  We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
 |-  ( A  e.  om  ->  (  _E  We  A  /\  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) ) )
 
Theoremonunsnss 6581 Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
 |-  ( ( B  e.  V  /\  ( A  u.  { B } )  e. 
 On )  ->  B  C_  A )
 
Theoremunfiexmid 6582* If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)
 |-  ( ( x  e. 
 Fin  /\  y  e.  Fin )  ->  ( x  u.  y )  e.  Fin )   =>    |-  ( ph  \/  -.  ph )
 
Theoremunsnfi 6583 Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  V  /\  -.  B  e.  A ) 
 ->  ( A  u.  { B } )  e.  Fin )
 
Theoremunsnfidcex 6584 The  B  e.  V condition in unsnfi 6583. This is intended to show that unsnfi 6583 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  -.  B  e.  A  /\  ( A  u.  { B } )  e.  Fin )  -> DECID  -.  B  e.  _V )
 
Theoremunsnfidcel 6585 The  -.  B  e.  A condition in unsnfi 6583. This is intended to show that unsnfi 6583 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  V  /\  ( A  u.  { B } )  e.  Fin )  -> DECID  -.  B  e.  A )
 
Theoremunfidisj 6586 The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( A  u.  B )  e. 
 Fin )
 
Theoremundifdcss 6587* Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
 |-  ( A  =  ( B  u.  ( A 
 \  B ) )  <-> 
 ( B  C_  A  /\  A. x  e.  A DECID  x  e.  B ) )
 
Theoremundifdc 6588* Union of complementary parts into whole. This is a case where we can strengthen undifss 3350 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
 |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  B  e.  Fin  /\  B  C_  A )  ->  A  =  ( B  u.  ( A  \  B ) ) )
 
Theoremundiffi 6589 Union of complementary parts into whole. This is a case where we can strengthen undifss 3350 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  A  =  ( B  u.  ( A  \  B ) ) )
 
Theoremunfiin 6590 The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  e.  Fin )  ->  ( A  u.  B )  e.  Fin )
 
Theoremprfidisj 6591 A pair is finite if it consists of two unequal sets. For the case where  A  =  B, see snfig 6485. For the cases where one or both is a proper class, see prprc1 3535, prprc2 3536, or prprc 3537. (Contributed by Jim Kingdon, 31-May-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B ) 
 ->  { A ,  B }  e.  Fin )
 
Theoremtpfidisj 6592 A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  A  =/=  C )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  { A ,  B ,  C }  e.  Fin )
 
Theoremxpfi 6593 The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (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 )
 
Theorem3xpfi 6594 The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( V  e.  Fin  ->  ( ( V  X.  V )  X.  V )  e.  Fin )
 
Theoremfisseneq 6595 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 )
 
Theoremssfirab 6596* A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A. x  e.  A DECID  ps )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  e.  Fin )
 
Theoremssfidc 6597* A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
 |-  ( ( A  e.  Fin  /\  B  C_  A  /\  A. x  e.  A DECID  x  e.  B )  ->  B  e.  Fin )
 
Theoremsnon0 6598 An ordinal which is a singleton is  { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
 |-  ( ( A  e.  V  /\  { A }  e.  On )  ->  A  =  (/) )
 
Theoremfnfi 6599 A version of fnex 5482 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremfundmfi 6600 The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  Fun  A )  ->  dom  A  e.  Fin )
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