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	dedth3v 4001	Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4000. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
|- ( A = if ( ph , A , D ) -> ( ps <-> ch ) )   &    |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )   &    |- ta   =>    |- ( ph -> ps )
Theorem	dedth4v 4002	Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4000. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
|- ( A = if ( ph , A , R ) -> ( ps <-> ch ) )   &    |- ( B = if ( ph , B , S ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , C , T ) -> ( th <-> ta ) )   &    |- ( D = if ( ph , D , U ) -> ( ta <-> et ) )   &    |- et   =>    |- ( ph -> ps )
Theorem	elimhyp 4003	Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 3996. (Contributed by NM, 15-May-1999.)
|- ( A = if ( ph , A , B ) -> ( ph <-> ps ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> ps ) )   &    |- ch   =>    |- ps
Theorem	elimhyp2v 4004	Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
|- ( A = if ( ph , A , C ) -> ( ph <-> ch ) )   &    |- ( B = if ( ph , B , D ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , A , C ) -> ( ta <-> et ) )   &    |- ( D = if ( ph , B , D ) -> ( et <-> th ) )   &    |- ta   =>    |- th
Theorem	elimhyp3v 4005	Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
|- ( A = if ( ph , A , D ) -> ( ph <-> ch ) )   &    |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )   &    |- ( D = if ( ph , A , D ) -> ( et <-> ze ) )   &    |- ( R = if ( ph , B , R ) -> ( ze <-> si ) )   &    |- ( S = if ( ph , C , S ) -> ( si <-> ta ) )   &    |- et   =>    |- ta
Theorem	elimhyp4v 4006	Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3996). (Contributed by NM, 16-Apr-2005.)
|- ( A = if ( ph , A , D ) -> ( ph <-> ch ) )   &    |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )   &    |- ( F = if ( ph , F , G ) -> ( ta <-> ps ) )   &    |- ( D = if ( ph , A , D ) -> ( et <-> ze ) )   &    |- ( R = if ( ph , B , R ) -> ( ze <-> si ) )   &    |- ( S = if ( ph , C , S ) -> ( si <-> rh ) )   &    |- ( G = if ( ph , F , G ) -> ( rh <-> ps ) )   &    |- et   =>    |- ps
Theorem	elimel 4007	Eliminate a membership hypothesis for weak deduction theorem, when special case B e. C is provable. (Contributed by NM, 15-May-1999.)
|- B e. C   =>    |- if ( A e. C , A , B ) e. C
Theorem	elimdhyp 4008	Version of elimhyp 4003 where the hypothesis is deduced from the final antecedent. See ghomgrplem 29204 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
|- ( ph -> ps )   &    |- ( A = if ( ph , A , B ) -> ( ps <-> ch ) )   &    |- ( B = if ( ph , A , B ) -> ( th <-> ch ) )   &    |- th   =>    |- ch
Theorem	keephyp 4009	Transform a hypothesis ps that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   &    |- ps   &    |- ch   =>    |- th
Theorem	keephyp2v 4010	Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3996). (Contributed by NM, 16-Apr-2005.)
|- ( A = if ( ph , A , C ) -> ( ps <-> ch ) )   &    |- ( B = if ( ph , B , D ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , A , C ) -> ( ta <-> et ) )   &    |- ( D = if ( ph , B , D ) -> ( et <-> th ) )   &    |- ps   &    |- ta   =>    |- th
Theorem	keephyp3v 4011	Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
|- ( A = if ( ph , A , D ) -> ( rh <-> ch ) )   &    |- ( B = if ( ph , B , R ) -> ( ch <-> th ) )   &    |- ( C = if ( ph , C , S ) -> ( th <-> ta ) )   &    |- ( D = if ( ph , A , D ) -> ( et <-> ze ) )   &    |- ( R = if ( ph , B , R ) -> ( ze <-> si ) )   &    |- ( S = if ( ph , C , S ) -> ( si <-> ta ) )   &    |- rh   &    |- et   =>    |- ta
Theorem	keepel 4012	Keep a membership hypothesis for weak deduction theorem, when special case B e. C is provable. (Contributed by NM, 14-Aug-1999.)
|- A e. C   &    |- B e. C   =>    |- if ( ph , A , B ) e. C
Theorem	ifex 4013	Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
|- A e. _V   &    |- B e. _V   =>    |- if ( ph , A , B ) e. _V
Theorem	ifexg 4014	Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. _V )
2.1.16  Power classes
Syntax	cpw 4015	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 4016*	Soundness justification theorem for df-pw 4017. (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 4017*	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 = { 3 , 5 , 7 }, then ~P A = { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } } (ex-pw 25276). We will later introduce the Axiom of Power Sets ax-pow 4634, which can be expressed in class notation per pwexg 4640. Still later we will prove, in hashpw 12497, 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, 24-Jun-1993.)
|- ~P A = { x | x C_ A }
Theorem	pweq 4018	Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)
|- ( A = B -> ~P A = ~P B )
Theorem	pweqi 4019	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
|- A = B   =>    |- ~P A = ~P B
Theorem	pweqd 4020	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ~P A = ~P B )
Theorem	elpw 4021	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	selpw 4022*	Setvar variable membership in a power class (common case). See elpw 4021. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( x e. ~P A <-> x C_ A )
Theorem	elpwg 4023	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4619. (Contributed by NM, 6-Aug-2000.)
|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )
Theorem	elpwi 4024	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
|- ( A e. ~P B -> A C_ B )
Theorem	elpwid 4025	An element of a power class is a subclass. Deduction form of elpwi 4024. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ~P B )   =>    |- ( ph -> A C_ B )
Theorem	elelpwi 4026	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 4027	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 4028	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 4029	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 4030*	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
Theorem	snjust 4031*	Soundness justification theorem for df-sn 4033. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x = A } = { y | y = A }
Syntax	csn 4032	Extend class notation to include singleton.
class { A }
Definition	df-sn 4033*	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 4045. (Contributed by NM, 21-Jun-1993.)
|- { A } = { x | x = A }
Syntax	cpr 4034	Extend class notation to include unordered pair.
class { A , B }
Definition	df-pr 4035	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, A e. { 1 , -u 1 } -> ( A ^ 2 ) = 1 (ex-pr 25277). They are unordered, so { A , B } = { B , A } as proven by prcom 4110. For a more traditional definition, but requiring a dummy variable, see dfpr2 4047. (Contributed by NM, 21-Jun-1993.)
|- { A , B } = ( { A } u. { B } )
Syntax	ctp 4036	Extend class notation to include unordered triplet.
class { A , B , C }
Definition	df-tp 4037	Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
|- { A , B , C } = ( { A , B } u. { C } )
Syntax	cop 4038	Extend class notation to include ordered pair.
class <. A , B >.
Definition	df-op 4039*	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 4242, opprc2 4243, and 0nelop 4746). For Kuratowski's actual definition when the arguments are sets, see dfop 4218. For the justifying theorem (for sets) see opth 4730. See dfopif 4216 for an equivalent formulation using the if operation. 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 4039 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4039 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 } } }, justified by opthwiener 4758. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition <. A , B >._3 = { A , { A , B } } is justified by opthreg 8052, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is <. A , B >._4 = ( ( A X. { (/) } ) u. ( B X. { { (/) } } ) ), justified by opthprc 5056. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 12352. An ordered pair of real numbers can also be represented by a complex number as shown by cru 10548. Kuratowski's ordered pair definition is standard for ZFC set theory, but it is very inconvenient to use in New Foundations theory because it is not type-level; a common alternative definition in New Foundations is the Definition from [Rosser] p. 281. Since there are other ways to define ordered pairs, we discourage direct use of this definition so that most theorems won't depend on this particular construction; theorems will instead rely on dfopif 4216. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
|- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
Syntax	cotp 4040	Extend class notation to include ordered triple.
class <. A , B , C >.
Definition	df-ot 4041	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 4042	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 4043	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 4044	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 4045	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsn 4046*	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	dfpr2 4047*	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 4048	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	eldifpr 4049	Membership in a set with two elements removed. Similar to eldifsn 4157 and eldiftp 4075. (Contributed by Mario Carneiro, 18-Jul-2017.)
|- ( A e. ( B \ { C , D } ) <-> ( A e. B /\ A =/= C /\ A =/= D ) )
Theorem	elpr 4050	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 4051	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 4052	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 4053	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	nelprd 4054	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	elsncg 4055	There is only 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	elsnc 4056	There is only 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	elsni 4057	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	snidg 4058	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 4059	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 4060	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	ssnid 4061	A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- x e. { x }
Theorem	elsnc2g 4062	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	elsnc2 4063	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	ralsnsg 4064*	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	rexsns 4065*	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	rexsnsOLD 4066*	Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 4065 instead. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A e. V -> ( E. x e. { A } ph <-> [. A / x ]. ph ) )
Theorem	ralsng 4067*	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 4068*	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	2ralsng 4069*	Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A } A. y e. { B } ph <-> ch ) )
Theorem	exsnrex 4070	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 4071*	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 4072*	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 4073	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	eltpg 4074	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 4075	Membership in a set with three elements removed. Similar to eldifsn 4157 and eldifpr 4049. (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 4076	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 4077	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 4078*	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 4079	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 4080	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 4081*	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 4082*	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 4083*	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 4084*	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 4085*	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 4086*	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 4087*	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 4088*	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 4089	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   =>    |- F/_ x { A }
Theorem	csbsng 4090	Distribute proper substitution through the singleton of a class. csbsng 4090 is derived from the virtual deduction proof csbsngVD 33794. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
Theorem	csbprg 4091	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 4092	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 4093	Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)
|- ( A =/= B -> ( { A } i^i { B } ) = (/) )
Theorem	disjpr2 4094	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 4095	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 4096	A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
|- ( A e. _V <-> { A } =/= (/) )
Theorem	r19.12sn 4097*	Special case of r19.12 2983 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	r19.12snOLD 4098*	Older form of r19.12sn 4097. Obsolete as of 18-Mar-2020. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.) (New usage is discouraged.)
|- A e. _V   =>    |- ( E. x e. { A } A. y e. B ph <-> A. y e. B E. x e. { A } ph )
Theorem	rabsn 4099*	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 4100*	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 } , (/) )