P. 41 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ralsn 4001*	Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. { A } ph <-> ps )
Theorem	rexsn 4002*	Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. { A } ph <-> ps )
Theorem	elpwunsn 4003	Membership in an extension of a power class. (Contributed by NM, 26-Mar-2007.)
|- ( A e. ( ~P ( B u. { C } ) \ ~P B ) -> C e. A )
Theorem	eqoreldif 4004	An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.)
|- ( B e. C -> ( A e. C <-> ( A = B \/ A e. ( C \ { B } ) ) ) )
Theorem	eltpg 4005	Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
|- ( A e. V -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )
Theorem	eldiftp 4006	Membership in a set with three elements removed. Similar to eldifsn 4088 and eldifpr 3982. (Contributed by David A. Wheeler, 22-Jul-2017.)
|- ( A e. ( B \ { C , D , E } ) <-> ( A e. B /\ ( A =/= C /\ A =/= D /\ A =/= E ) ) )
Theorem	eltpi 4007	A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
|- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) )
Theorem	eltp 4008	A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- A e. _V   =>    |- ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) )
Theorem	dftp2 4009*	Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
|- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
Theorem	nfpr 4010	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x { A , B }
Theorem	ifpr 4011	Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
|- ( ( A e. C /\ B e. D ) -> if ( ph , A , B ) e. { A , B } )
Theorem	ralprg 4012*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )
Theorem	rexprg 4013*	Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
Theorem	raltpg 4014*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )
Theorem	rextpg 4015*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) )
Theorem	ralpr 4016*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) )
Theorem	rexpr 4017*	Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) )
Theorem	raltp 4018*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) )
Theorem	rextp 4019*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) )
Theorem	nfsn 4020	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   =>    |- F/_ x { A }
Theorem	csbsng 4021	Distribute proper substitution through the singleton of a class. csbsng 4021 is derived from the virtual deduction proof csbsngVD 37353. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
Theorem	csbprg 4022	Distribute proper substitution through a pair of classes. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
|- ( C e. V -> [_ C / x ]_ { A , B } = { [_ C / x ]_ A , [_ C / x ]_ B } )
Theorem	disjsn 4023	Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
|- ( ( A i^i { B } ) = (/) <-> -. B e. A )
Theorem	disjsn2 4024	Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
|- ( A =/= B -> ( { A } i^i { B } ) = (/) )
Theorem	disjpr2 4025	The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
|- ( ( ( A =/= C /\ B =/= C ) /\ ( A =/= D /\ B =/= D ) ) -> ( { A , B } i^i { C , D } ) = (/) )
Theorem	disjprsn 4026	The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)
|- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
Theorem	snprc 4027	The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 21-Jun-1993.)
|- ( -. A e. _V <-> { A } = (/) )
Theorem	snnzb 4028	A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
|- ( A e. _V <-> { A } =/= (/) )
Theorem	r19.12sn 4029*	Special case of r19.12 2903 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 18-Mar-2020.)
|- ( A e. V -> ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) )
Theorem	rabsn 4030*	Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)
|- ( B e. A -> { x e. A | x = B } = { B } )
Theorem	rabsnifsb 4031*	A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)
|- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
Theorem	rabsnif 4032*	A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- { x e. { A } | ph } = if ( ps , { A } , (/) )
Theorem	rabrsn 4033*	A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)
|- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )
Theorem	euabsn2 4034*	Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( E! x ph <-> E. y { x | ph } = { y } )
Theorem	euabsn 4035	Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)
|- ( E! x ph <-> E. x { x | ph } = { x } )
Theorem	reusn 4036*	A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- ( E! x e. A ph <-> E. y { x e. A | ph } = { y } )
Theorem	absneu 4037	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
|- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph )
Theorem	rabsneu 4038	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( ( A e. V /\ { x e. B | ph } = { A } ) -> E! x e. B ph )
Theorem	eusn 4039*	Two ways to express " A is a singleton." (Contributed by NM, 30-Oct-2010.)
|- ( E! x x e. A <-> E. x A = { x } )
Theorem	rabsnt 4040*	Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
|- B e. _V   &    |- ( x = B -> ( ph <-> ps ) )   =>    |- ( { x e. A | ph } = { B } -> ps )
Theorem	prcom 4041	Commutative law for unordered pairs. (Contributed by NM, 15-Jul-1993.)
|- { A , B } = { B , A }
Theorem	preq1 4042	Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
|- ( A = B -> { A , C } = { B , C } )
Theorem	preq2 4043	Equality theorem for unordered pairs. (Contributed by NM, 15-Jul-1993.)
|- ( A = B -> { C , A } = { C , B } )
Theorem	preq12 4044	Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
Theorem	preq1i 4045	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- A = B   =>    |- { A , C } = { B , C }
Theorem	preq2i 4046	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- A = B   =>    |- { C , A } = { C , B }
Theorem	preq12i 4047	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- A = B   &    |- C = D   =>    |- { A , C } = { B , D }
Theorem	preq1d 4048	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ph -> A = B )   =>    |- ( ph -> { A , C } = { B , C } )
Theorem	preq2d 4049	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ph -> A = B )   =>    |- ( ph -> { C , A } = { C , B } )
Theorem	preq12d 4050	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> { A , C } = { B , D } )
Theorem	tpeq1 4051	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
|- ( A = B -> { A , C , D } = { B , C , D } )
Theorem	tpeq2 4052	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
|- ( A = B -> { C , A , D } = { C , B , D } )
Theorem	tpeq3 4053	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
|- ( A = B -> { C , D , A } = { C , D , B } )
Theorem	tpeq1d 4054	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> { A , C , D } = { B , C , D } )
Theorem	tpeq2d 4055	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> { C , A , D } = { C , B , D } )
Theorem	tpeq3d 4056	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> { C , D , A } = { C , D , B } )
Theorem	tpeq123d 4057	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   &    |- ( ph -> E = F )   =>    |- ( ph -> { A , C , E } = { B , D , F } )
Theorem	tprot 4058	Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
|- { A , B , C } = { B , C , A }
Theorem	tpcoma 4059	Swap 1st and 2nd members of an unordered triple. (Contributed by NM, 22-May-2015.)
|- { A , B , C } = { B , A , C }
Theorem	tpcomb 4060	Swap 2nd and 3rd members of an unordered triple. (Contributed by NM, 22-May-2015.)
|- { A , B , C } = { A , C , B }
Theorem	tpass 4061	Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- { A , B , C } = ( { A } u. { B , C } )
Theorem	qdass 4062	Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- ( { A , B } u. { C , D } ) = ( { A , B , C } u. { D } )
Theorem	qdassr 4063	Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
|- ( { A , B } u. { C , D } ) = ( { A } u. { B , C , D } )
Theorem	tpidm12 4064	Unordered triple { A , A , B } is just an overlong way to write { A , B }. (Contributed by David A. Wheeler, 10-May-2015.)
|- { A , A , B } = { A , B }
Theorem	tpidm13 4065	Unordered triple { A , B , A } is just an overlong way to write { A , B }. (Contributed by David A. Wheeler, 10-May-2015.)
|- { A , B , A } = { A , B }
Theorem	tpidm23 4066	Unordered triple { A , B , B } is just an overlong way to write { A , B }. (Contributed by David A. Wheeler, 10-May-2015.)
|- { A , B , B } = { A , B }
Theorem	tpidm 4067	Unordered triple { A , A , A } is just an overlong way to write { A }. (Contributed by David A. Wheeler, 10-May-2015.)
|- { A , A , A } = { A }
Theorem	tppreq3 4068	An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
|- ( B = C -> { A , B , C } = { A , B } )
Theorem	prid1g 4069	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
|- ( A e. V -> A e. { A , B } )
Theorem	prid2g 4070	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
|- ( B e. V -> B e. { A , B } )
Theorem	prid1 4071	An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 24-Jun-1993.)
|- A e. _V   =>    |- A e. { A , B }
Theorem	prid2 4072	An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
|- B e. _V   =>    |- B e. { A , B }
Theorem	prprc1 4073	A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
|- ( -. A e. _V -> { A , B } = { B } )
Theorem	prprc2 4074	A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
|- ( -. B e. _V -> { A , B } = { A } )
Theorem	prprc 4075	An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
|- ( ( -. A e. _V /\ -. B e. _V ) -> { A , B } = (/) )
Theorem	tpid1 4076	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- A e. _V   =>    |- A e. { A , B , C }
Theorem	tpid2 4077	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- B e. _V   =>    |- B e. { A , B , C }
Theorem	tpid3g 4078	Closed theorem form of tpid3 4079. This proof was automatically generated from the virtual deduction proof tpid3gVD 37301 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
|- ( A e. B -> A e. { C , D , A } )
Theorem	tpid3 4079	One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- C e. _V   =>    |- C e. { A , B , C }
Theorem	snnzg 4080	The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
|- ( A e. V -> { A } =/= (/) )
Theorem	snnz 4081	The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
|- A e. _V   =>    |- { A } =/= (/)
Theorem	prnz 4082	A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
|- A e. _V   =>    |- { A , B } =/= (/)
Theorem	prnzg 4083	A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
|- ( A e. V -> { A , B } =/= (/) )
Theorem	tpnz 4084	A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
|- A e. _V   =>    |- { A , B , C } =/= (/)
Theorem	tpnzd 4085	A triplet containing a set is not empty. (Contributed by Thierry Arnoux, 8-Apr-2019.)
|- ( ph -> A e. V )   =>    |- ( ph -> { A , B , C } =/= (/) )
Theorem	raltpd 4086*	Convert a quantification over a triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )   &    |- ( ( ph /\ x = B ) -> ( ps <-> th ) )   &    |- ( ( ph /\ x = C ) -> ( ps <-> ta ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   =>    |- ( ph -> ( A. x e. { A , B , C } ps <-> ( ch /\ th /\ ta ) ) )
Theorem	snss 4087	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.)
|- A e. _V   =>    |- ( A e. B <-> { A } C_ B )
Theorem	eldifsn 4088	Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
|- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) )
Theorem	eldifsni 4089	Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
|- ( A e. ( B \ { C } ) -> A =/= C )
Theorem	neldifsn 4090	A is not in ( B \ { A } ). (Contributed by David Moews, 1-May-2017.)
|- -. A e. ( B \ { A } )
Theorem	neldifsnd 4091	A is not in ( B \ { A } ). Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> -. A e. ( B \ { A } ) )
Theorem	rexdifsn 4092	Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
|- ( E. x e. ( A \ { B } ) ph <-> E. x e. A ( x =/= B /\ ph ) )
Theorem	raldifsni 4093	Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
|- ( A. x e. ( A \ { B } ) -. ph <-> A. x e. A ( ph -> x = B ) )
Theorem	raldifsnb 4094*	Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
|- ( A. x e. A ( x =/= Y -> ph ) <-> A. x e. ( A \ { Y } ) ph )
Theorem	eldifvsn 4095	A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)
|- ( A e. V -> ( A e. ( _V \ { B } ) <-> A =/= B ) )
Theorem	snssg 4096	The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
Theorem	difsn 4097	An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( -. A e. B -> ( B \ { A } ) = B )
Theorem	difprsnss 4098	Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( { A , B } \ { A } ) C_ { B }
Theorem	difprsn1 4099	Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
|- ( A =/= B -> ( { A , B } \ { A } ) = { B } )
Theorem	difprsn2 4100	Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
|- ( A =/= B -> ( { A , B } \ { B } ) = { A } )