P. 38 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-pr 3701	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so { A , B } = { B , A } as proven by prcom 3772. For a more traditional definition, but requiring a dummy variable, see dfpr2 3713. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = ( { A } u. { B } )
Definition	df-tp 3702	Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
|- { A , B , C } = ( { A , B } u. { C } )
Definition	df-op 3703*	Definition of an ordered pair, equivalent to Kuratowski's definition { { A } , { A , B } } when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3910 and opprc2 3911). For Kuratowski's actual definition when the arguments are sets, see dfop 3887. Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as <. A , B >. = { { A } , { A , B } }, which has different behavior from our df-op 3703 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3703 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses. There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition <. A , B >.2 = { { { A } , (/) } , { { B } } }. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is <. A , B >.3 = { A , { A , B } }, but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
Definition	df-ot 3704	Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
|- <. A , B , C >. = <. <. A , B >. , C >.
Theorem	sneq 3705	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 3706	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 3707	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 3708	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsng 3709	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( A e. { B } <-> A = B ) )
Theorem	elsn 3710	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
|- A e. _V   =>    |- ( A e. { B } <-> A = B )
Theorem	velsn 3711	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
|- ( x e. { A } <-> x = A )
Theorem	elsni 3712	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	dfpr2 3713*	Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A , B } = { x | ( x = A \/ x = B ) }
Theorem	elprg 3714	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
Theorem	elpr 3715	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
|- A e. _V   =>    |- ( A e. { B , C } <-> ( A = B \/ A = C ) )
Theorem	elpr2 3716	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
|- B e. _V   &    |- C e. _V   =>    |- ( A e. { B , C } <-> ( A = B \/ A = C ) )
Theorem	elpri 3717	If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
|- ( A e. { B , C } -> ( A = B \/ A = C ) )
Theorem	nelpri 3718	If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
|- A =/= B   &    |- A =/= C   =>    |- -. A e. { B , C }
Theorem	prneli 3719	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using e/. (Contributed by David A. Wheeler, 10-May-2015.)
|- A =/= B   &    |- A =/= C   =>    |- A e/ { B , C }
Theorem	nelprd 3720	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( ph -> A =/= B )   &    |- ( ph -> A =/= C )   =>    |- ( ph -> -. A e. { B , C } )
Theorem	eldifpr 3721	Membership in a set with two elements removed. Similar to eldifsn 3825 and eldiftp 3740. (Contributed by Mario Carneiro, 18-Jul-2017.)
|- ( A e. ( B \ { C , D } ) <-> ( A e. B /\ A =/= C /\ A =/= D ) )
Theorem	rexdifpr 3722	Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
|- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )
Theorem	snidg 3723	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
|- ( A e. V -> A e. { A } )
Theorem	snidb 3724	A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
|- ( A e. _V <-> A e. { A } )
Theorem	snid 3725	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
|- A e. _V   =>    |- A e. { A }
Theorem	vsnid 3726	A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- x e. { x }
Theorem	elsn2g 3727	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)
|- ( B e. V -> ( A e. { B } <-> A = B ) )
Theorem	elsn2 3728	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 12-Jun-1994.)
|- B e. _V   =>    |- ( A e. { B } <-> A = B )
Theorem	nelsn 3729	If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)
|- ( A =/= B -> -. A e. { B } )
Theorem	mosn 3730*	A singleton has at most one element. This works whether A is a proper class or not, and in that sense can be seen as encompassing both snmg 3815 and snprc 3759. (Contributed by Jim Kingdon, 30-Aug-2018.)
|- E* x x e. { A }
Theorem	ralsnsg 3731*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
Theorem	ralsns 3732*	Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
Theorem	rexsns 3733*	Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
|- ( E. x e. { A } ph <-> [. A / x ]. ph )
Theorem	ralsng 3734*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x e. { A } ph <-> ps ) )
Theorem	rexsng 3735*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
Theorem	exsnrex 3736	There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
|- ( E. x M = { x } <-> E. x e. M M = { x } )
Theorem	ralsn 3737*	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 3738*	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	eltpg 3739	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 3740	Membership in a set with three elements removed. Similar to eldifsn 3825 and eldifpr 3721. (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 3741	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 3742	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 3743*	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 3744	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x { A , B }
Theorem	ralprg 3745*	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 3746*	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 3747*	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 3748*	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 3749*	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 3750*	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 3751*	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 3752*	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	sbcsng 3753*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( A e. V -> ( [. A / x ]. ph <-> A. x e. { A } ph ) )
Theorem	nfsn 3754	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   =>    |- F/_ x { A }
Theorem	csbsng 3755	Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
Theorem	disjsn 3756	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 3757	Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
|- ( A =/= B -> ( { A } i^i { B } ) = (/) )
Theorem	disjpr2 3758	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	snprc 3759	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, 5-Aug-1993.)
|- ( -. A e. _V <-> { A } = (/) )
Theorem	r19.12sn 3760*	Special case of r19.12 2651 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (Revised by BJ, 20-Dec-2021.)
|- ( A e. V -> ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph ) )
Theorem	rabsn 3761*	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 3762*	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 3763*	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	rabrsndc 3764*	A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
|- A e. _V   &    |- DECID ph   =>    |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )
Theorem	euabsn2 3765*	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 3766	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 3767*	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 3768	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
|- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph )
Theorem	rabsneu 3769	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 3770*	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 3771*	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 3772	Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = { B , A }
Theorem	preq1 3773	Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
|- ( A = B -> { A , C } = { B , C } )
Theorem	preq2 3774	Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { C , A } = { C , B } )
Theorem	preq12 3775	Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )
Theorem	preq1i 3776	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- A = B   =>    |- { A , C } = { B , C }
Theorem	preq2i 3777	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- A = B   =>    |- { C , A } = { C , B }
Theorem	preq12i 3778	Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- A = B   &    |- C = D   =>    |- { A , C } = { B , D }
Theorem	preq1d 3779	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ph -> A = B )   =>    |- ( ph -> { A , C } = { B , C } )
Theorem	preq2d 3780	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ph -> A = B )   =>    |- ( ph -> { C , A } = { C , B } )
Theorem	preq12d 3781	Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> { A , C } = { B , D } )
Theorem	tpeq1 3782	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
|- ( A = B -> { A , C , D } = { B , C , D } )
Theorem	tpeq2 3783	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
|- ( A = B -> { C , A , D } = { C , B , D } )
Theorem	tpeq3 3784	Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)
|- ( A = B -> { C , D , A } = { C , D , B } )
Theorem	tpeq1d 3785	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> { A , C , D } = { B , C , D } )
Theorem	tpeq2d 3786	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> { C , A , D } = { C , B , D } )
Theorem	tpeq3d 3787	Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
|- ( ph -> A = B )   =>    |- ( ph -> { C , D , A } = { C , D , B } )
Theorem	tpeq123d 3788	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 3789	Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
|- { A , B , C } = { B , C , A }
Theorem	tpcoma 3790	Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
|- { A , B , C } = { B , A , C }
Theorem	tpcomb 3791	Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)
|- { A , B , C } = { A , C , B }
Theorem	tpass 3792	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 3793	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 3794	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 3795	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 3796	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 3797	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 3798	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 3799	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 3800	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 } )