P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	en1uniel 6801	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
|- ( S ~~ 1o -> U. S e. S )
Theorem	2dom 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 )
Theorem	fundmen 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 )
Theorem	fundmeng 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 )
Theorem	cnven 6805	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ( Rel A /\ A e. V ) -> A ~~ `' A )
Theorem	cnvct 6806	If a set is dominated by om, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> `' A ~<_ om )
Theorem	fndmeng 6807	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( F Fn A /\ A e. C ) -> A ~~ F )
Theorem	mapsnen 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
Theorem	map1 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 )
Theorem	en2sn 6810	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
|- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } )
Theorem	snfig 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 )
Theorem	fiprc 6812	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
|- Fin e/ _V
Theorem	unen 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 ) )
Theorem	enpr2d 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 )
Theorem	ssct 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 )
Theorem	1domsn 6816	A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
|- { A } ~<_ 1o
Theorem	enm 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 )
Theorem	xpsnen 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
Theorem	xpsneng 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 )
Theorem	xp1en 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 )
Theorem	endisj 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 ) = (/) )
Theorem	xpcomf1o 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 )
Theorem	xpcomco 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 )
Theorem	xpcomen 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 )
Theorem	xpcomeng 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 ) )
Theorem	xpsnen2g 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 )
Theorem	xpassen 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 ) )
Theorem	xpdom2 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 ) )
Theorem	xpdom2g 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 ) )
Theorem	xpdom1g 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 ) )
Theorem	xpdom3m 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 ) )
Theorem	xpdom1 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 ) )
Theorem	fopwdom 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 )
Theorem	0domg 6834	Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A e. V -> (/) ~<_ A )
Theorem	dom0 6835	A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
|- ( A ~<_ (/) <-> A = (/) )
Theorem	0dom 6836	Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- A e. _V   =>    |- (/) ~<_ A
Theorem	enen1 6837	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) )
Theorem	enen2 6838	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) )
Theorem	domen1 6839	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )
Theorem	domen2 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.)
Theorem	xpf1o 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 ) )
Theorem	xpen 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 ) )
Theorem	mapen 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 ) )
Theorem	mapdom1g 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 ) )
Theorem	mapxpen 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 ) ) )
Theorem	xpmapenlem 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 ) )
Theorem	xpmapen 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 ) )
Theorem	ssenen 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
Theorem	phplem1 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 } ) )
Theorem	phplem2 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 } ) )
Theorem	phplem3 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 } ) )
Theorem	phplem4 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 ) )
Theorem	phplem3g 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 } ) )
Theorem	nneneq 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 ) )
Theorem	php5 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 )
Theorem	snnen2og 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 )
Theorem	snnen2oprc 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 )
Theorem	1nen2 6858	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
|- -. 1o ~~ 2o
Theorem	phplem4dom 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 ) )
Theorem	php5dom 6860	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
Theorem	nndomo 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 ) )
Theorem	phpm 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 )
Theorem	phpelm 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 )
Theorem	phplem4on 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
Theorem	fict 6865	A finite set is dominated by om. Also see finct 7112. (Contributed by Thierry Arnoux, 27-Mar-2018.)
|- ( A e. Fin -> A ~<_ om )
Theorem	fidceq 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 )
Theorem	fidifsnen 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 } ) )
Theorem	fidifsnid 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 )
Theorem	nnfi 6869	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- ( A e. om -> A e. Fin )
Theorem	enfi 6870	Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
|- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) )
Theorem	enfii 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 )
Theorem	ssfilem 6872*	Lemma for ssfiexmid 6873. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- { z e. { (/) } | ph } e. Fin   =>    |- ( ph \/ -. ph )
Theorem	ssfiexmid 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 )
Theorem	infiexmid 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 )
Theorem	domfiexmid 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 )
Theorem	dif1en 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 )
Theorem	dif1enen 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 } ) )
Theorem	fiunsnnn 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 )
Theorem	php5fin 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 } ) )
Theorem	fisbth 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 )
Theorem	0fin 6881	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
|- (/) e. Fin
Theorem	fin0 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 ) )
Theorem	fin0or 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 ) )
Theorem	diffitest 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 )
Theorem	findcard 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 )
Theorem	findcard2 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 )
Theorem	findcard2s 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 )
Theorem	findcard2d 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 )
Theorem	findcard2sd 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 )
Theorem	diffisn 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 )
Theorem	diffifi 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 )
Theorem	infnfi 6892	An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
|- ( om ~<_ A -> -. A e. Fin )
Theorem	ominf 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
Theorem	isinfinf 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 ) )
Theorem	ac6sfi 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 ) )
Theorem	tridc 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 )
Theorem	fimax2gtrilemstep 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 )
Theorem	fimax2gtri 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 )
Theorem	finexdc 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 )
Theorem	dfrex2fin 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 ) )