P. 71 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	entr 7001	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
|- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C )
Theorem	domtr 7002	Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ( ( A ~<_ B /\ B ~<_ C ) -> A ~<_ C )
Theorem	entri 7003	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- A ~~ C
Theorem	entr2i 7004	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- C ~~ A
Theorem	entr3i 7005	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- A ~~ C   =>    |- B ~~ C
Theorem	entr4i 7006	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- C ~~ B   =>    |- A ~~ C
Theorem	endomtr 7007	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C )
Theorem	domentr 7008	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C )
Theorem	f1imaeng 7009	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( F : A -1-1-> B /\ C C_ A /\ C e. V ) -> ( F " C ) ~~ C )
Theorem	f1imaen2g 7010	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7011 does not need ax-setind 4641.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
|- ( ( ( F : A -1-1-> B /\ B e. V ) /\ ( C C_ A /\ C e. V ) ) -> ( F " C ) ~~ C )
Theorem	f1imaen 7011	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
|- C e. _V   =>    |- ( ( F : A -1-1-> B /\ C C_ A ) -> ( F " C ) ~~ C )
Theorem	en0 7012	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
|- ( A ~~ (/) <-> A = (/) )
Theorem	ensn1 7013	A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
|- A e. _V   =>    |- { A } ~~ 1o
Theorem	ensn1g 7014	A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> { A } ~~ 1o )
Theorem	enpr1g 7015	{ A , A } has only one element. (Contributed by FL, 15-Feb-2010.)
|- ( A e. V -> { A , A } ~~ 1o )
Theorem	en1 7016*	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
|- ( A ~~ 1o <-> E. x A = { x } )
Theorem	en1bg 7017	A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
|- ( A e. V -> ( A ~~ 1o <-> A = { U. A } ) )
Theorem	reuen1 7018*	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E! x e. A ph <-> { x e. A | ph } ~~ 1o )
Theorem	euen1 7019	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E! x ph <-> { x | ph } ~~ 1o )
Theorem	euen1b 7020*	Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
|- ( A ~~ 1o <-> E! x x e. A )
Theorem	en1uniel 7021	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
|- ( S ~~ 1o -> U. S e. S )
Theorem	en1m 7022*	A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 1o -> E. x x e. A )
Theorem	2dom 7023*	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 7024	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 7025	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 7026	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ( Rel A /\ A e. V ) -> A ~~ `' A )
Theorem	cnvct 7027	If a set is dominated by om, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> `' A ~<_ om )
Theorem	fndmeng 7028	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( F Fn A /\ A e. C ) -> A ~~ F )
Theorem	mapsnen 7029	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 7030	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 7031	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
|- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } )
Theorem	snfig 7032	A singleton is finite. For the proper class case, see snprc 3738. (Contributed by Jim Kingdon, 13-Apr-2020.)
|- ( A e. V -> { A } e. Fin )
Theorem	fiprc 7033	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
|- Fin e/ _V
Theorem	unen 7034	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 7035	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 7036	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 7037	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E* x ph <-> { x | ph } ~<_ 1o )
Theorem	modom2 7038*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E* x x e. A <-> A ~<_ 1o )
Theorem	rex2dom 7039*	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 7040	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 7041*	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 7042*	A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 2o -> E. x x e. A )
Theorem	ssct 7043	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 7044	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 7045*	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 7046	A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
|- ( ( A e. V /\ B e. A ) -> 1o ~<_ A )
Theorem	enm 7047*	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 7048	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 7049	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 7050	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 7051*	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 7052*	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 7053*	Composition with the bijection of xpcomf1o 7052 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 7054	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 7055	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 7056	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 7057	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 7058	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 7059	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 7060	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 7061*	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 7062	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 7063*	Lemma for pw2f1odc 7064. (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 7064*	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 7065	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 7066	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 7067	A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
|- ( A ~<_ (/) <-> A = (/) )
Theorem	0dom 7068	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 7069	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( A ~~ C <-> B ~~ C ) )
Theorem	enen2 7070	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
|- ( A ~~ B -> ( C ~~ A <-> C ~~ B ) )
Theorem	domen1 7071	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
|- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )
Theorem	domen2 7072	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 7073*	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 7074	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 7075	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 7076	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 7077	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 7078*	Lemma for xpmapen 7079. (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 7079	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 7080*	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 7081	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 7082	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 7083	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 7085. (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 7084	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 7085	A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 7083 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. suc A ) -> A ~~ ( suc A \ { B } ) )
Theorem	nneneq 7086	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 7087	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 7088	A singleton { A } is never equinumerous with the ordinal number 2. If A is a proper class, see snnen2oprc 7089. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. V -> -. { A } ~~ 2o )
Theorem	snnen2oprc 7089	A singleton { A } is never equinumerous with the ordinal number 2. If A is a set, see snnen2og 7088. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( -. A e. _V -> -. { A } ~~ 2o )
Theorem	1nen2 7090	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
|- -. 1o ~~ 2o
Theorem	phplem4dom 7091	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 7092	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
Theorem	nndomo 7093	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 7094	Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- -. 2o ~<_ 1o
Theorem	phpm 7095*	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 7081 through phplem4 7084, nneneq 7086, 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 7096	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 7097	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 7098	A finite set is dominated by om. Also see finct 7358. (Contributed by Thierry Arnoux, 27-Mar-2018.)
|- ( A e. Fin -> A ~<_ om )
Theorem	fidceq 7099	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 7100	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 } ) )