P. 37 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ifmdc 3601	If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
|- ( A e. if ( ph , B , C ) -> DECID ph )
Theorem	ifnetruedc 3602	Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )
Theorem	ifnefals 3603	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )
Theorem	ifnebibdc 3604	The converse of ifbi 3581 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) )
2.1.16  Power classes
Syntax	cpw 3605	Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class ~P A
Theorem	pwjust 3606*	Soundness justification theorem for df-pw 3607. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x C_ A } = { y | y C_ A }
Definition	df-pw 3607*	Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of _V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if A is { 3 , 5 , 7 }, then ~P A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)
|- ~P A = { x | x C_ A }
Theorem	pweq 3608	Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ~P A = ~P B )
Theorem	pweqi 3609	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
|- A = B   =>    |- ~P A = ~P B
Theorem	pweqd 3610	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ~P A = ~P B )
Theorem	elpw 3611	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
|- A e. _V   =>    |- ( A e. ~P B <-> A C_ B )
Theorem	velpw 3612*	Setvar variable membership in a power class (common case). See elpw 3611. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( x e. ~P A <-> x C_ A )
Theorem	elpwg 3613	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )
Theorem	elpwi 3614	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
|- ( A e. ~P B -> A C_ B )
Theorem	elpwb 3615	Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
|- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )
Theorem	elpwid 3616	An element of a power class is a subclass. Deduction form of elpwi 3614. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ~P B )   =>    |- ( ph -> A C_ B )
Theorem	elelpwi 3617	If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)
|- ( ( A e. B /\ B e. ~P C ) -> A e. C )
Theorem	nfpw 3618	Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- F/_ x A   =>    |- F/_ x ~P A
Theorem	pwidg 3619	Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( A e. V -> A e. ~P A )
Theorem	pwid 3620	A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
|- A e. _V   =>    |- A e. ~P A
Theorem	pwss 3621*	Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
|- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) )
2.1.17  Unordered and ordered pairs
Syntax	csn 3622	Extend class notation to include singleton.
class { A }
Syntax	cpr 3623	Extend class notation to include unordered pair.
class { A , B }
Syntax	ctp 3624	Extend class notation to include unordered triplet.
class { A , B , C }
Syntax	cop 3625	Extend class notation to include ordered pair.
class <. A , B >.
Syntax	cotp 3626	Extend class notation to include ordered triple.
class <. A , B , C >.
Theorem	snjust 3627*	Soundness justification theorem for df-sn 3628. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x = A } = { y | y = A }
Definition	df-sn 3628*	Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of _V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3636. (Contributed by NM, 5-Aug-1993.)
|- { A } = { x | x = A }
Definition	df-pr 3629	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so { A , B } = { B , A } as proven by prcom 3698. For a more traditional definition, but requiring a dummy variable, see dfpr2 3641. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = ( { A } u. { B } )
Definition	df-tp 3630	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 3631*	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 3830 and opprc2 3831). For Kuratowski's actual definition when the arguments are sets, see dfop 3807. 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 3631 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3631 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 3632	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 3633	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 3634	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 3635	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 3636	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsng 3637	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 3638	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 3639	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 3640	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	dfpr2 3641*	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 3642	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 3643	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 3644	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 3645	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 3646	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 3647	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 3648	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 3649	Membership in a set with two elements removed. Similar to eldifsn 3749 and eldiftp 3668. (Contributed by Mario Carneiro, 18-Jul-2017.)
|- ( A e. ( B \ { C , D } ) <-> ( A e. B /\ A =/= C /\ A =/= D ) )
Theorem	rexdifpr 3650	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 3651	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 3652	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 3653	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 3654	A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- x e. { x }
Theorem	elsn2g 3655	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 3656	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 3657	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 3658*	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 3740 and snprc 3687. (Contributed by Jim Kingdon, 30-Aug-2018.)
|- E* x x e. { A }
Theorem	ralsnsg 3659*	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 3660*	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 3661*	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 3662*	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 3663*	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 3664	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 3665*	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 3666*	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 3667	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 3668	Membership in a set with three elements removed. Similar to eldifsn 3749 and eldifpr 3649. (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 3669	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 3670	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 3671*	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 3672	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 3673*	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 3674*	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 3675*	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 3676*	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 3677*	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 3678*	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 3679*	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 3680*	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 3681*	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 3682	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   =>    |- F/_ x { A }
Theorem	csbsng 3683	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 3684	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 3685	Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
|- ( A =/= B -> ( { A } i^i { B } ) = (/) )
Theorem	disjpr2 3686	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 3687	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 3688*	Special case of r19.12 2603 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 3689*	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	rabrsndc 3690*	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 3691*	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 3692	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 3693*	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 3694	Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)
|- ( ( A e. V /\ { x | ph } = { A } ) -> E! x ph )
Theorem	rabsneu 3695	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 3696*	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 3697*	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 3698	Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = { B , A }
Theorem	preq1 3699	Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)
|- ( A = B -> { A , C } = { B , C } )
Theorem	preq2 3700	Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { C , A } = { C , B } )