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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremen1uniel 6801 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( S  ~~  1o  ->  U. S  e.  S )
 
Theorem2dom 6802* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
 |-  ( 2o  ~<_  A  ->  E. x  e.  A  E. y  e.  A  -.  x  =  y )
 
Theoremfundmen 6803 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  F  e.  _V   =>    |-  ( Fun  F  ->  dom  F  ~~  F )
 
Theoremfundmeng 6804 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
 |-  ( ( F  e.  V  /\  Fun  F )  ->  dom  F  ~~  F )
 
Theoremcnven 6805 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( ( Rel  A  /\  A  e.  V ) 
 ->  A  ~~  `' A )
 
Theoremcnvct 6806 If a set is dominated by  om, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A  ~<_  om  ->  `' A  ~<_  om )
 
Theoremfndmeng 6807 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( F  Fn  A  /\  A  e.  C )  ->  A  ~~  F )
 
Theoremmapsnen 6808 Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  ^m  { B } )  ~~  A
 
Theoremmap1 6809 Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)
 |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
 
Theoremen2sn 6810 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  { A }  ~~  { B } )
 
Theoremsnfig 6811 A singleton is finite. For the proper class case, see snprc 3657. (Contributed by Jim Kingdon, 13-Apr-2020.)
 |-  ( A  e.  V  ->  { A }  e.  Fin )
 
Theoremfiprc 6812 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
 |- 
 Fin  e/  _V
 
Theoremunen 6813 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A 
 ~~  B  /\  C  ~~  D )  /\  (
 ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( A  u.  C )  ~~  ( B  u.  D ) )
 
Theoremenpr2d 6814 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  { A ,  B }  ~~  2o )
 
Theoremssct 6815 A subset of a set dominated by 
om is dominated by 
om. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theorem1domsn 6816 A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
 |- 
 { A }  ~<_  1o
 
Theoremenm 6817* A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
 |-  ( ( A  ~~  B  /\  E. x  x  e.  A )  ->  E. y  y  e.  B )
 
Theoremxpsnen 6818 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  { B } )  ~~  A
 
Theoremxpsneng 6819 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  { B } )  ~~  A )
 
Theoremxp1en 6820 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  V  ->  ( A  X.  1o )  ~~  A )
 
Theoremendisj 6821* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 E. x E. y
 ( ( x  ~~  A  /\  y  ~~  B )  /\  ( x  i^i  y )  =  (/) )
 
Theoremxpcomf1o 6822* The canonical bijection from  ( A  X.  B
) to  ( B  X.  A ). (Contributed by Mario Carneiro, 23-Apr-2014.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   =>    |-  F : ( A  X.  B ) -1-1-onto-> ( B  X.  A )
 
Theoremxpcomco 6823* Composition with the bijection of xpcomf1o 6822 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
 |-  F  =  ( x  e.  ( A  X.  B )  |->  U. `' { x } )   &    |-  G  =  ( y  e.  B ,  z  e.  A  |->  C )   =>    |-  ( G  o.  F )  =  ( z  e.  A ,  y  e.  B  |->  C )
 
Theoremxpcomen 6824 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  X.  B )  ~~  ( B  X.  A )
 
Theoremxpcomeng 6825 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
 
Theoremxpsnen2g 6826 A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A }  X.  B )  ~~  B )
 
Theoremxpassen 6827 Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A  X.  B )  X.  C )  ~~  ( A  X.  ( B  X.  C ) )
 
Theoremxpdom2 6828 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 6829 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 6830 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 6831* 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 6832 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 6833 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 6834 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( A  e.  V  -> 
 (/)  ~<_  A )
 
Theoremdom0 6835 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
 |-  ( A  ~<_  (/)  <->  A  =  (/) )
 
Theorem0dom 6836 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   =>    |-  (/)  ~<_  A
 
Theoremenen1 6837 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( A  ~~  C  <->  B 
 ~~  C ) )
 
Theoremenen2 6838 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
 |-  ( A  ~~  B  ->  ( C  ~~  A  <->  C 
 ~~  B ) )
 
Theoremdomen1 6839 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( A  ~<_  C  <->  B  ~<_  C )
 )
 
Theoremdomen2 6840 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
 |-  ( A  ~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B )
 )
 
2.6.29  Equinumerosity (cont.)
 
Theoremxpf1o 6841* 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 6842 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 6843 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 6844 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 6845 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 6846* Lemma for xpmapen 6847. (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 6847 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 6848* 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 ) }
 )
 
2.6.30  Pigeonhole Principle
 
Theoremphplem1 6849 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 6850 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 6851 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 6853. (Contributed by NM, 26-May-1998.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremphplem4 6852 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 6853 A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6851 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( ( A  e.  om 
 /\  B  e.  suc  A )  ->  A  ~~  ( suc  A  \  { B } ) )
 
Theoremnneneq 6854 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 6855 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 6856 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a proper class, see snnen2oprc 6857. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  V  ->  -.  { A }  ~~  2o )
 
Theoremsnnen2oprc 6857 A singleton  { A } is never equinumerous with the ordinal number 2. If  A is a set, see snnen2og 6856. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( -.  A  e.  _V 
 ->  -.  { A }  ~~  2o )
 
Theorem1nen2 6858 One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
 |- 
 -.  1o  ~~  2o
 
Theoremphplem4dom 6859 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 6860 A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
 |-  ( A  e.  om  ->  -.  suc  A  ~<_  A )
 
Theoremnndomo 6861 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 6862* 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 6849 through phplem4 6852, nneneq 6854, 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 6863 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 6864 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 ) )
 
2.6.31  Finite sets
 
Theoremfict 6865 A finite set is dominated by  om. Also see finct 7112. (Contributed by Thierry Arnoux, 27-Mar-2018.)
 |-  ( A  e.  Fin  ->  A 
 ~<_  om )
 
Theoremfidceq 6866 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 6867 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 6868 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 3738 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 6869 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( A  e.  om  ->  A  e.  Fin )
 
Theoremenfi 6870 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
 |-  ( A  ~~  B  ->  ( A  e.  Fin  <->  B  e.  Fin ) )
 
Theoremenfii 6871 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 6872* Lemma for ssfiexmid 6873. (Contributed by Jim Kingdon, 3-Feb-2022.)
 |- 
 { z  e.  { (/)
 }  |  ph }  e.  Fin   =>    |-  ( ph  \/  -.  ph )
 
Theoremssfiexmid 6873* 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 6874* 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 6875* 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 6876 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 6877 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 6878 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 6879 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 6880 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 6881 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
 |-  (/)  e.  Fin
 
Theoremfin0 6882* 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 6883* 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 6884* 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 6885* 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 6886* 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 6887* Variation of findcard2 6886 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 6888* Deduction version of findcard2 6886. If you also need  y  e.  Fin (which doesn't come for free due to ssfiexmid 6873), use findcard2sd 6889 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 6889* 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 6890 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 6891 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 6892 An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
 |-  ( om  ~<_  A  ->  -.  A  e.  Fin )
 
Theoremominf 6893 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 6764. (Contributed by NM, 2-Jun-1998.)
 |- 
 -.  om  e.  Fin
 
Theoremisinfinf 6894* 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 6895* 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 6896* 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 6897* Lemma for fimax2gtri 6898. 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 6898* 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 6899* 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 6900* 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 )
 )
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