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	map1 7101	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 7102	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
|- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } )
Theorem	snfig 7103	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 7104	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
|- Fin e/ _V
Theorem	unen 7105	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 7106	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 7107	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 7108	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E* x ph <-> { x | ph } ~<_ 1o )
Theorem	modom2 7109*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E* x x e. A <-> A ~<_ 1o )
Theorem	rex2dom 7110*	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 7111	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 7112*	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 7113*	A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 2o -> E. x x e. A )
Theorem	ssct 7114	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 7115	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 7116*	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 7117	A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
|- ( ( A e. V /\ B e. A ) -> 1o ~<_ A )
Theorem	enm 7118*	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 7119	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 7120	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 7121	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 7122*	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 7123*	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 7124*	Composition with the bijection of xpcomf1o 7123 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 7125	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 7126	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 7127	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 7128	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 7129	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 7130	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 7131	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 7132*	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 7133	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 7134*	Lemma for pw2f1odc 7135. (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 7135*	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 7136	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 7137	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 7138	A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
|- ( A ~<_ (/) <-> A = (/) )
Theorem	0dom 7139	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 7140	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) )
Theorem	enen2 7141	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) )
Theorem	domen1 7142	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )
Theorem	domen2 7143	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 7144*	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 7145	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 7146	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 7147	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 7148	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 7149*	Lemma for xpmapen 7150. (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 7150	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 7151	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 7152*	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 7153	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 7154	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 7155	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 7157. (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 7156	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 7157	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 7155 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )
Theorem	nneneq 7158	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 7159	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 7160	A singleton { A } is never equinumerous with the ordinal number 2. If A is a proper class, see snnen2oprc 7161. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. V -> -. { A } ~~ 2o )
Theorem	snnen2oprc 7161	A singleton { A } is never equinumerous with the ordinal number 2. If A is a set, see snnen2og 7160. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( -. A e. _V -> -. { A } ~~ 2o )
Theorem	1nen2 7162	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
|- -. 1o ~~ 2o
Theorem	phplem4dom 7163	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 7164	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
Theorem	nndomo 7165	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 7166	Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- -. 2o ~<_ 1o
Theorem	phpm 7167*	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 7153 through phplem4 7156, nneneq 7158, 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 7168	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 7169	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 7170	A finite set is dominated by om. Also see finct 7456. (Contributed by Thierry Arnoux, 27-Mar-2018.)
|- ( A e. Fin -> A ~<_ om )
Theorem	fidceq 7171	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 7172	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 7173	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 7174	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- ( A e. om -> A e. Fin )
Theorem	enfi 7175	Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
|- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) )
Theorem	enfii 7176	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 7177*	Lemma for ssfiexmid 7178. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- { z e. { (/) } | ph } e. Fin   =>    |- ( ph \/ -. ph )
Theorem	ssfiexmid 7178*	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 7179*	Lemma for ssfiexmidt 7180. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- ( ph -> { z e. { (/) } | ps } e. Fin )   =>    |- ( ph -> ( ps \/ -. ps ) )
Theorem	ssfiexmidt 7180*	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 7181*	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 7182*	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 7183	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 7184	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 7185	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 7186	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 7187	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 7188	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
|- (/) e. Fin
Theorem	fin0 7189*	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 7190*	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 7191*	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 7192*	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 7193*	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 7194*	Variation of findcard2 7193 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 7195*	Deduction version of findcard2 7193. If you also need y e. Fin (which doesn't come for free due to ssfiexmid 7178), use findcard2sd 7196 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 7196*	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 7197	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 7198	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 7199	An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
|- ( om ~<_ A -> -. A e. Fin )
Theorem	ominf 7200	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 7053. (Contributed by NM, 2-Jun-1998.)
|- -. om e. Fin