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	raaanv 3601*	Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
|- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )
Theorem	sbss 3602*	Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|- ( [ y / x ] x C_ A <-> y C_ A )
Theorem	sbcssg 3603	Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
|- ( A e. V -> ( [. A / x ]. B C_ C <-> [_ A / x ]_ B C_ [_ A / x ]_ C ) )
Theorem	dcun 3604	The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) (Revised by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> DECID C e. A )   &    |- ( ph -> DECID C e. B )   =>    |- ( ph -> DECID C e. ( A u. B ) )
2.1.15  Conditional operator
Syntax	cif 3605	Extend class notation to include the conditional operator. See df-if 3606 for a description. (In older databases this was denoted "ded".)
class if ( ph , A , B )
Definition	df-if 3606*	Define the conditional operator. Read if ( ph , A , B ) as "if ph then A else B". See iftrue 3610 and iffalse 3613 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise." In the absence of excluded middle, this will tend to be useful where ph is decidable (in the sense of df-dc 842). (Contributed by NM, 15-May-1999.)
|- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
Theorem	dfif6 3607*	An alternate definition of the conditional operator df-if 3606 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
|- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )
Theorem	ifeq1 3608	Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
Theorem	ifeq2 3609	Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A = B -> if ( ph , C , A ) = if ( ph , C , B ) )
Theorem	iftrue 3610	Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> if ( ph , A , B ) = A )
Theorem	iftruei 3611	Inference associated with iftrue 3610. (Contributed by BJ, 7-Oct-2018.)
|- ph   =>    |- if ( ph , A , B ) = A
Theorem	iftrued 3612	Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ch )   =>    |- ( ph -> if ( ch , A , B ) = A )
Theorem	iffalse 3613	Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
|- ( -. ph -> if ( ph , A , B ) = B )
Theorem	iffalsei 3614	Inference associated with iffalse 3613. (Contributed by BJ, 7-Oct-2018.)
|- -. ph   =>    |- if ( ph , A , B ) = B
Theorem	iffalsed 3615	Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> -. ch )   =>    |- ( ph -> if ( ch , A , B ) = B )
Theorem	ifnefalse 3616	When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse 3613 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
|- ( A =/= B -> if ( A = B , C , D ) = D )
Theorem	elif 3617	Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
|- ( A e. if ( ph , B , C ) <-> ( ( ph /\ A e. B ) \/ ( -. ph /\ A e. C ) ) )
Theorem	ifsbdc 3618	Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
|- ( if ( ph , A , B ) = A -> C = D )   &    |- ( if ( ph , A , B ) = B -> C = E )   =>    |- (DECID ph -> C = if ( ph , D , E ) )
Theorem	dfif3 3619*	Alternate definition of the conditional operator df-if 3606. Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- C = { x | ph }   =>    |- if ( ph , A , B ) = ( ( A i^i C ) u. ( B i^i ( _V \ C ) ) )
Theorem	ifssun 3620	A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , A , B ) C_ ( A u. B )
Theorem	ifidss 3621	A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , A , A ) C_ A
Theorem	ifeq12 3622	Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
|- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )
Theorem	ifeq1d 3623	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifeq2d 3624	Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )
Theorem	ifeq12d 3625	Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , D ) )
Theorem	ifbi 3626	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
|- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )
Theorem	ifbid 3627	Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> if ( ps , A , B ) = if ( ch , A , B ) )
Theorem	ifbieq1d 3628	Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = B )   =>    |- ( ph -> if ( ps , A , C ) = if ( ch , B , C ) )
Theorem	ifbieq2i 3629	Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph <-> ps )   &    |- A = B   =>    |- if ( ph , C , A ) = if ( ps , C , B )
Theorem	ifbieq2d 3630	Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = B )   =>    |- ( ph -> if ( ps , C , A ) = if ( ch , C , B ) )
Theorem	ifbieq12i 3631	Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
|- ( ph <-> ps )   &    |- A = C   &    |- B = D   =>    |- if ( ph , A , B ) = if ( ps , C , D )
Theorem	ifbieq12d 3632	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )
Theorem	nfifd 3633	Deduction version of nfif 3634. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ph -> F/ x ps )   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x if ( ps , A , B ) )
Theorem	nfif 3634	Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/_ x if ( ph , A , B )
Theorem	ifcldadc 3635	Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
|- ( ( ph /\ ps ) -> A e. C )   &    |- ( ( ph /\ -. ps ) -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifeq1dadc 3636	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifeq2dadc 3637	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ -. ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )
Theorem	ifeqdadc 3638	Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
|- ( ( ph /\ ps ) -> A = C )   &    |- ( ( ph /\ -. ps ) -> B = C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) = C )
Theorem	ifbothdadc 3639	A formula th containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   &    |- ( ( et /\ ph ) -> ps )   &    |- ( ( et /\ -. ph ) -> ch )   &    |- ( et -> DECID ph )   =>    |- ( et -> th )
Theorem	ifbothdc 3640	A wff th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   =>    |- ( ( ps /\ ch /\ DECID ph ) -> th )
Theorem	ifiddc 3641	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
|- (DECID ph -> if ( ph , A , A ) = A )
Theorem	eqifdc 3642	Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
|- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
Theorem	ifcldcd 3643	Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifnotdc 3644	Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
|- (DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
Theorem	2if2dc 3645	Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
|- ( ( ph /\ ps ) -> D = A )   &    |- ( ( ph /\ -. ps /\ th ) -> D = B )   &    |- ( ( ph /\ -. ps /\ -. th ) -> D = C )   &    |- ( ph -> DECID ps )   &    |- ( ( ph /\ -. ps ) -> DECID th )   =>    |- ( ph -> D = if ( ps , A , if ( th , B , C ) ) )
Theorem	ifandc 3646	Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- (DECID ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )
Theorem	ifordc 3647	Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- (DECID ph -> if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) ) )
Theorem	ifmdc 3648	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 3649	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 3650	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )
Theorem	ifnebibdc 3651	The converse of ifbi 3626 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 3652	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 3653*	Soundness justification theorem for df-pw 3654. (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 3654*	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 3655	Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ~P A = ~P B )
Theorem	pweqi 3656	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
|- A = B   =>    |- ~P A = ~P B
Theorem	pweqd 3657	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ~P A = ~P B )
Theorem	elpw 3658	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 3659*	Setvar variable membership in a power class (common case). See elpw 3658. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( x e. ~P A <-> x C_ A )
Theorem	elpwg 3660	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 3661	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
|- ( A e. ~P B -> A C_ B )
Theorem	elpwb 3662	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 3663	An element of a power class is a subclass. Deduction form of elpwi 3661. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ~P B )   =>    |- ( ph -> A C_ B )
Theorem	elelpwi 3664	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 3665	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 3666	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 3667	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 3668*	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 3669	Extend class notation to include singleton.
class { A }
Syntax	cpr 3670	Extend class notation to include unordered pair.
class { A , B }
Syntax	ctp 3671	Extend class notation to include unordered triplet.
class { A , B , C }
Syntax	cop 3672	Extend class notation to include ordered pair.
class <. A , B >.
Syntax	cotp 3673	Extend class notation to include ordered triple.
class <. A , B , C >.
Theorem	snjust 3674*	Soundness justification theorem for df-sn 3675. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x = A } = { y | y = A }
Definition	df-sn 3675*	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 3683. (Contributed by NM, 5-Aug-1993.)
|- { A } = { x | x = A }
Definition	df-pr 3676	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so { A , B } = { B , A } as proven by prcom 3747. For a more traditional definition, but requiring a dummy variable, see dfpr2 3688. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = ( { A } u. { B } )
Definition	df-tp 3677	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 3678*	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 3884 and opprc2 3885). For Kuratowski's actual definition when the arguments are sets, see dfop 3861. 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 3678 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3678 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 3679	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 3680	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 3681	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 3682	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 3683	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsng 3684	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 3685	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 3686	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 3687	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	dfpr2 3688*	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 3689	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 3690	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 3691	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 3692	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 3693	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 3694	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 3695	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 3696	Membership in a set with two elements removed. Similar to eldifsn 3800 and eldiftp 3715. (Contributed by Mario Carneiro, 18-Jul-2017.)
|- ( A e. ( B \ { C , D } ) <-> ( A e. B /\ A =/= C /\ A =/= D ) )
Theorem	rexdifpr 3697	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 3698	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 3699	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 3700	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 }