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	0elixp 7001	Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.)
|- (/) e. X_ x e. (/) A
Theorem	ixpm 7002*	If an infinite Cartesian product of a family B ( x ) is inhabited, every B ( x ) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
|- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
Theorem	ixp0 7003	The infinite Cartesian product of a family B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
|- ( E. x e. A B = (/) -> X_ x e. A B = (/) )
Theorem	ixpssmap 7004*	An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
|- B e. _V   =>    |- X_ x e. A B C_ ( U_ x e. A B ^m A )
Theorem	resixp 7005*	Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
|- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. X_ x e. B C )
Theorem	mptelixpg 7006*	Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)
|- ( I e. V -> ( ( x e. I |-> J ) e. X_ x e. I K <-> A. x e. I J e. K ) )
Theorem	elixpsn 7007*	Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- ( A e. V -> ( F e. X_ x e. { A } B <-> E. y e. B F = { <. A , y >. } ) )
Theorem	ixpsnf1o 7008*	A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. A |-> ( { I } X. { x } ) )   =>    |- ( I e. V -> F : A -1-1-onto-> X_ y e. { I } A )
Theorem	mapsnf1o 7009*	A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- F = ( x e. A |-> ( { I } X. { x } ) )   =>    |- ( ( A e. V /\ I e. W ) -> F : A -1-1-onto-> ( A ^m { I } ) )
2.6.29  Equinumerosity
Syntax	cen 7010	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ~~
Syntax	cdom 7011	Extend class definition to include the dominance relation (curly less-than-or-equal)
class ~<_
Syntax	cfn 7012	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 7013*	Define the equinumerosity relation. Definition of [Enderton] p. 129. We define ~~ to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 7020. (Contributed by NM, 28-Mar-1998.)
|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
Definition	df-dom 7014*	Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 7024 and domen 7025. (Contributed by NM, 28-Mar-1998.)
|- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
Definition	df-fin 7015*	Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our " a e. Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 16916. (Contributed by NM, 22-Aug-2008.)
|- Fin = { x | E. y e. om x ~~ y }
Theorem	relen 7016	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~~
Theorem	reldom 7017	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~<_
Theorem	encv 7018	If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
|- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
Theorem	breng 7019*	Equinumerosity relation. This variation of bren 7020 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 7020. (Revised by BTernaryTau, 23-Sep-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
Theorem	bren 7020*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
Theorem	brdom2g 7021*	Dominance relation. This variation of brdomg 7022 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 7022. (Revised by BTernaryTau, 29-Nov-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomg 7022*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomi 7023*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~<_ B -> E. f f : A -1-1-> B )
Theorem	brdom 7024*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. f f : A -1-1-> B )
Theorem	domen 7025*	Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) )
Theorem	domeng 7026*	Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
|- ( B e. C -> ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) ) )
Theorem	ctex 7027	A class dominated by om is a set. See also ctfoex 7448 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
|- ( A ~<_ om -> A e. _V )
Theorem	f1oen4g 7028	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 7033 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1dom4g 7029	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 7034 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> A ~<_ B )
Theorem	f1oen3g 7030	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7033 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( ( F e. V /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1oen2g 7031	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7033 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1dom2g 7032	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 7034 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( A e. V /\ B e. W /\ F : A -1-1-> B ) -> A ~<_ B )
Theorem	f1oeng 7033	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
|- ( ( A e. C /\ F : A -1-1-onto-> B ) -> A ~~ B )
Theorem	f1domg 7034	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
|- ( B e. C -> ( F : A -1-1-> B -> A ~<_ B ) )
Theorem	f1oen 7035	The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
|- A e. _V   =>    |- ( F : A -1-1-onto-> B -> A ~~ B )
Theorem	f1dom 7036	The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
|- B e. _V   =>    |- ( F : A -1-1-> B -> A ~<_ B )
Theorem	isfi 7037*	Express " A is finite". Definition 10.29 of [TakeutiZaring] p. 91 (whose " Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
|- ( A e. Fin <-> E. x e. om A ~~ x )
Theorem	enssdom 7038	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
|- ~~ C_ ~<_
Theorem	endom 7039	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
|- ( A ~~ B -> A ~<_ B )
Theorem	enrefg 7040	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A e. V -> A ~~ A )
Theorem	enref 7041	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A e. _V   =>    |- A ~~ A
Theorem	eqeng 7042	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
|- ( A e. V -> ( A = B -> A ~~ B ) )
Theorem	domrefg 7043	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
|- ( A e. V -> A ~<_ A )
Theorem	en2d 7044*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> ( x e. A -> C e. X ) )   &    |- ( ph -> ( y e. B -> D e. Y ) )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> A ~~ B )
Theorem	en3d 7045*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( y e. B -> D e. A ) )   &    |- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )   =>    |- ( ph -> A ~~ B )
Theorem	en2i 7046*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
|- A e. _V   &    |- B e. _V   &    |- ( x e. A -> C e. _V )   &    |- ( y e. B -> D e. _V )   &    |- ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) )   =>    |- A ~~ B
Theorem	en3i 7047*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
|- A e. _V   &    |- B e. _V   &    |- ( x e. A -> C e. B )   &    |- ( y e. B -> D e. A )   &    |- ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) )   =>    |- A ~~ B
Theorem	dom2lem 7048*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
|- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )   =>    |- ( ph -> ( x e. A |-> C ) : A -1-1-> B )
Theorem	dom2d 7049*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)
|- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )   =>    |- ( ph -> ( B e. R -> A ~<_ B ) )
Theorem	dom3d 7050*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
|- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> A ~<_ B )
Theorem	dom2 7051*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
|- ( x e. A -> C e. B )   &    |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )   =>    |- ( B e. V -> A ~<_ B )
Theorem	dom3 7052*	A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C ( x ) and D ( y ), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
|- ( x e. A -> C e. B )   &    |- ( ( x e. A /\ y e. A ) -> ( C = D <-> x = y ) )   =>    |- ( ( A e. V /\ B e. W ) -> A ~<_ B )
Theorem	idssen 7053	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- _I C_ ~~
Theorem	domssr 7054	If C is a superset of B and B dominates A, then C also dominates A. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C )
Theorem	ssdomg 7055	A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( B e. V -> ( A C_ B -> A ~<_ B ) )
Theorem	ener 7056	Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ~~ Er _V
Theorem	ensymb 7057	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B <-> B ~~ A )
Theorem	ensym 7058	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B -> B ~~ A )
Theorem	ensymi 7059	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   =>    |- B ~~ A
Theorem	ensymd 7060	Symmetry of equinumerosity. Deduction form of ensym 7058. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A ~~ B )   =>    |- ( ph -> B ~~ A )
Theorem	entr 7061	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
|- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C )
Theorem	domtr 7062	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 7063	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- A ~~ C
Theorem	entr2i 7064	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- C ~~ A
Theorem	entr3i 7065	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- A ~~ C   =>    |- B ~~ C
Theorem	entr4i 7066	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- C ~~ B   =>    |- A ~~ C
Theorem	endomtr 7067	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C )
Theorem	domentr 7068	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C )
Theorem	f1imaeng 7069	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 7070	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7071 does not need ax-setind 4679.) (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 7071	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 7072	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
|- ( A ~~ (/) <-> A = (/) )
Theorem	ensn1 7073	A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
|- A e. _V   =>    |- { A } ~~ 1o
Theorem	ensn1g 7074	A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> { A } ~~ 1o )
Theorem	enpr1g 7075	{ A , A } has only one element. (Contributed by FL, 15-Feb-2010.)
|- ( A e. V -> { A , A } ~~ 1o )
Theorem	en1 7076*	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 7077	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 7078*	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 7079	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E! x ph <-> { x | ph } ~~ 1o )
Theorem	euen1b 7080*	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 7081	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
|- ( S ~~ 1o -> U. S e. S )
Theorem	en1m 7082*	A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 1o -> E. x x e. A )
Theorem	2dom 7083*	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 7084	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 7085	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 7086	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ( Rel A /\ A e. V ) -> A ~~ `' A )
Theorem	cnvct 7087	If a set is dominated by om, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> `' A ~<_ om )
Theorem	fndmeng 7088	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( F Fn A /\ A e. C ) -> A ~~ F )
Theorem	mapsnend 7089	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 7090	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 7091	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 7092	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
|- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } )
Theorem	snfig 7093	A singleton is finite. For the proper class case, see snprc 3770. (Contributed by Jim Kingdon, 13-Apr-2020.)
|- ( A e. V -> { A } e. Fin )
Theorem	fiprc 7094	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
|- Fin e/ _V
Theorem	unen 7095	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 7096	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 7097	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 7098	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E* x ph <-> { x | ph } ~<_ 1o )
Theorem	modom2 7099*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E* x x e. A <-> A ~<_ 1o )
Theorem	rex2dom 7100*	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 )