P. 70 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	resixp 6901*	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 6902*	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 6903*	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 6904*	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 6905*	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 } ) )
2.6.28  Equinumerosity
Syntax	cen 6906	Extend class definition to include the equinumerosity relation ("approximately equals" symbol)
class ~~
Syntax	cdom 6907	Extend class definition to include the dominance relation (curly less-than-or-equal)
class ~<_
Syntax	cfn 6908	Extend class definition to include the class of all finite sets.
class Fin
Definition	df-en 6909*	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 6916. (Contributed by NM, 28-Mar-1998.)
|- ~~ = { <. x , y >. | E. f f : x -1-1-onto-> y }
Definition	df-dom 6910*	Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6920 and domen 6921. (Contributed by NM, 28-Mar-1998.)
|- ~<_ = { <. x , y >. | E. f f : x -1-1-> y }
Definition	df-fin 6911*	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 16571. (Contributed by NM, 22-Aug-2008.)
|- Fin = { x | E. y e. om x ~~ y }
Theorem	relen 6912	Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~~
Theorem	reldom 6913	Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
|- Rel ~<_
Theorem	encv 6914	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 6915*	Equinumerosity relation. This variation of bren 6916 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6916. (Revised by BTernaryTau, 23-Sep-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
Theorem	bren 6916*	Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )
Theorem	brdom2g 6917*	Dominance relation. This variation of brdomg 6918 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6918. (Revised by BTernaryTau, 29-Nov-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomg 6918*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
Theorem	brdomi 6919*	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~<_ B -> E. f f : A -1-1-> B )
Theorem	brdom 6920*	Dominance relation. (Contributed by NM, 15-Jun-1998.)
|- B e. _V   =>    |- ( A ~<_ B <-> E. f f : A -1-1-> B )
Theorem	domen 6921*	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 6922*	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 6923	A class dominated by om is a set. See also ctfoex 7316 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 6924	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6929 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 6925	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 6930 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 6926	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6929 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 6927	The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6929 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 6928	The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6930 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 6929	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 6930	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 6931	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 6932	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 6933*	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 6934	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
|- ~~ C_ ~<_
Theorem	endom 6935	Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
|- ( A ~~ B -> A ~<_ B )
Theorem	enrefg 6936	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 6937	Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A e. _V   =>    |- A ~~ A
Theorem	eqeng 6938	Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
|- ( A e. V -> ( A = B -> A ~~ B ) )
Theorem	domrefg 6939	Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
|- ( A e. V -> A ~<_ A )
Theorem	en2d 6940*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ph -> ( x e. A -> C e. _V ) )   &    |- ( ph -> ( y e. B -> D e. _V ) )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> A ~~ B )
Theorem	en3d 6941*	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( 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 6942*	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 6943*	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 6944*	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 6945*	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 6946*	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 6947*	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 6948*	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 6949	Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- _I C_ ~~
Theorem	domssr 6950	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 6951	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 6952	Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
|- ~~ Er _V
Theorem	ensymb 6953	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
|- ( A ~~ B <-> B ~~ A )
Theorem	ensym 6954	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 6955	Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   =>    |- B ~~ A
Theorem	ensymd 6956	Symmetry of equinumerosity. Deduction form of ensym 6954. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A ~~ B )   =>    |- ( ph -> B ~~ A )
Theorem	entr 6957	Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
|- ( ( A ~~ B /\ B ~~ C ) -> A ~~ C )
Theorem	domtr 6958	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 6959	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- A ~~ C
Theorem	entr2i 6960	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- B ~~ C   =>    |- C ~~ A
Theorem	entr3i 6961	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- A ~~ C   =>    |- B ~~ C
Theorem	entr4i 6962	A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
|- A ~~ B   &    |- C ~~ B   =>    |- A ~~ C
Theorem	endomtr 6963	Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C )
Theorem	domentr 6964	Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
|- ( ( A ~<_ B /\ B ~~ C ) -> A ~<_ C )
Theorem	f1imaeng 6965	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 6966	A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6967 does not need ax-setind 4635.) (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 6967	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 6968	The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
|- ( A ~~ (/) <-> A = (/) )
Theorem	ensn1 6969	A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
|- A e. _V   =>    |- { A } ~~ 1o
Theorem	ensn1g 6970	A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
|- ( A e. V -> { A } ~~ 1o )
Theorem	enpr1g 6971	{ A , A } has only one element. (Contributed by FL, 15-Feb-2010.)
|- ( A e. V -> { A , A } ~~ 1o )
Theorem	en1 6972*	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 6973	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 6974*	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 6975	Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E! x ph <-> { x | ph } ~~ 1o )
Theorem	euen1b 6976*	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 6977	A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
|- ( S ~~ 1o -> U. S e. S )
Theorem	en1m 6978*	A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 1o -> E. x x e. A )
Theorem	2dom 6979*	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 6980	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 6981	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 6982	A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( ( Rel A /\ A e. V ) -> A ~~ `' A )
Theorem	cnvct 6983	If a set is dominated by om, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
|- ( A ~<_ om -> `' A ~<_ om )
Theorem	fndmeng 6984	A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ( F Fn A /\ A e. C ) -> A ~~ F )
Theorem	mapsnen 6985	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 6986	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 6987	Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
|- ( ( A e. C /\ B e. D ) -> { A } ~~ { B } )
Theorem	snfig 6988	A singleton is finite. For the proper class case, see snprc 3734. (Contributed by Jim Kingdon, 13-Apr-2020.)
|- ( A e. V -> { A } e. Fin )
Theorem	fiprc 6989	The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
|- Fin e/ _V
Theorem	unen 6990	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 6991	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 6992	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 6993	Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
|- ( E* x ph <-> { x | ph } ~<_ 1o )
Theorem	modom2 6994*	Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E* x x e. A <-> A ~<_ 1o )
Theorem	rex2dom 6995*	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 6996	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 6997*	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 6998*	A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 2o -> E. x x e. A )
Theorem	ssct 6999	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 7000	A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
|- { A } ~<_ 1o