P. 72 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fundmeng 7101	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 7102	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ( Rel A /\ A e. V ) -> A ~~ `' A )
Theorem	cnvct 7103	If a set is dominated by om, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> `' A ~<_ om )
Theorem	fndmeng 7104	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( F Fn A /\ A e. C ) -> A ~~ F )
Theorem	mapsnend 7105	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.) (Revised by Glauco Siliprandi, 24-Dec-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( A ^m { B } ) ~~ A )
Theorem	mapsnen 7106	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 7107	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 7108	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
|- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } )
Theorem	snfig 7109	A singleton is finite. For the proper class case, see snprc 3774. (Contributed by Jim Kingdon, 13-Apr-2020.)
|- ( A e. V -> { A } e. Fin )
Theorem	fiprc 7110	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
|- Fin e/ _V
Theorem	unen 7111	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	en2prd 7112	Two proper unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. Y )   &    |- ( ph -> A =/= B )   &    |- ( ph -> C =/= D )   =>    |- ( ph -> { A , B } ~~ { C , D } )
Theorem	1dom1el 7113	If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
|- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )
Theorem	modom 7114	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E* x ph <-> { x | ph } ~<_ 1o )
Theorem	modom2 7115*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E* x x e. A <-> A ~<_ 1o )
Theorem	rex2dom 7116*	A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)
|- ( ( A e. V /\ E. x e. A E. y e. A x =/= y ) -> 2o ~<_ A )
Theorem	enpr2d 7117	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	en2 7118*	A set equinumerous to ordinal 2 is an unordered pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- ( A ~~ 2o -> E. x E. y A = { x , y } )
Theorem	en2m 7119*	A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 2o -> E. x x e. A )
Theorem	ssct 7120	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 7121	A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
|- { A } ~<_ 1o
Theorem	dom1o 7122*	Two ways of saying that a set is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A e. V -> ( 1o ~<_ A <-> E. j j e. A ) )
Theorem	dom1oi 7123	A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
|- ( ( A e. V /\ B e. A ) -> 1o ~<_ A )
Theorem	enm 7124*	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 7125	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 7126	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 7127	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 7128*	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 7129*	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 7130*	Composition with the bijection of xpcomf1o 7129 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 7131	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 7132	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 7133	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 7134	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 7135	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 7136	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 7137	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 7138*	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 7139	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	pw2f1odclem 7140*	Lemma for pw2f1odc 7141. (Contributed by Mario Carneiro, 6-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. W )   &    |- ( ph -> B =/= C )   &    |- ( ph -> A. p e. A A. q e. ~P ADECID p e. q )   =>    |- ( ph -> ( ( S e. ~P A /\ G = ( z e. A |-> if ( z e. S , C , B ) ) ) <-> ( G e. ( { B , C } ^m A ) /\ S = ( `' G " { C } ) ) ) )
Theorem	pw2f1odc 7141*	The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. W )   &    |- ( ph -> B =/= C )   &    |- ( ph -> A. p e. A A. q e. ~P ADECID p e. q )   &    |- F = ( x e. ~P A |-> ( z e. A |-> if ( z e. x , C , B ) ) )   =>    |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) )
Theorem	fopwdom 7142	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 7143	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 7144	A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
|- ( A ~<_ (/) <-> A = (/) )
Theorem	0dom 7145	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 7146	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) )
Theorem	enen2 7147	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) )
Theorem	domen1 7148	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )
Theorem	domen2 7149	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) )
2.6.30  Equinumerosity (cont.)
Theorem	xpf1o 7150*	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 7151	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 7152	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 7153	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 7154	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 7155*	Lemma for xpmapen 7156. (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 7156	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	mapunen 7157	Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( ( ( A e. V /\ B e. W /\ C e. X ) /\ ( A i^i B ) = (/) ) -> ( C ^m ( A u. B ) ) ~~ ( ( C ^m A ) X. ( C ^m B ) ) )
Theorem	ssenen 7158*	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 ) } )
2.6.31  Pigeonhole Principle
Theorem	phplem1 7159	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 7160	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 7161	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 7163. (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 7162	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 7163	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 7161 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )
Theorem	nneneq 7164	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 7165	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 7166	A singleton { A } is never equinumerous with the ordinal number 2. If A is a proper class, see snnen2oprc 7167. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. V -> -. { A } ~~ 2o )
Theorem	snnen2oprc 7167	A singleton { A } is never equinumerous with the ordinal number 2. If A is a set, see snnen2og 7166. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( -. A e. _V -> -. { A } ~~ 2o )
Theorem	1nen2 7168	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
|- -. 1o ~~ 2o
Theorem	phplem4dom 7169	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 7170	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
Theorem	nndomo 7171	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	1ndom2 7172	Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- -. 2o ~<_ 1o
Theorem	phpm 7173*	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 7159 through phplem4 7162, nneneq 7164, 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 7174	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 7175	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 ) )
2.6.32  Finite sets
Theorem	fict 7176	A finite set is dominated by om. Also see finct 7462. (Contributed by Thierry Arnoux, 27-Mar-2018.)
|- ( A e. Fin -> A ~<_ om )
Theorem	fidceq 7177	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 7178	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 7179	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 3861 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 7180	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- ( A e. om -> A e. Fin )
Theorem	enfi 7181	Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
|- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) )
Theorem	enfii 7182	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 7183*	Lemma for ssfiexmid 7184. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- { z e. { (/) } | ph } e. Fin   =>    |- ( ph \/ -. ph )
Theorem	ssfiexmid 7184*	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	ssfilemd 7185*	Lemma for ssfiexmidt 7186. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- ( ph -> { z e. { (/) } | ps } e. Fin )   =>    |- ( ph -> ( ps \/ -. ps ) )
Theorem	ssfiexmidt 7186*	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 7187*	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 7188*	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 7189	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 7190	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 7191	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 7192	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 7193	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	0fi 7194	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
|- (/) e. Fin
Theorem	fin0 7195*	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 7196*	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 7197*	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 7198*	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 7199*	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 7200*	Variation of findcard2 7199 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 )