P. 36 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r19.28mv 3501*	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )
Theorem	r19.45mv 3502*	Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
|- ( E. x x e. A -> ( E. x e. A ( ph \/ ps ) <-> ( ph \/ E. x e. A ps ) ) )
Theorem	r19.44mv 3503*	Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
|- ( E. y y e. A -> ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ ps ) ) )
Theorem	r19.27m 3504*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- F/ x ps   =>    |- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )
Theorem	r19.27mv 3505*	Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- ( E. x x e. A -> ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ ps ) ) )
Theorem	rzal 3506*	Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A = (/) -> A. x e. A ph )
Theorem	rexn0 3507*	Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3508). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- ( E. x e. A ph -> A =/= (/) )
Theorem	rexm 3508*	Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
|- ( E. x e. A ph -> E. x x e. A )
Theorem	ralidm 3509*	Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
|- ( A. x e. A A. x e. A ph <-> A. x e. A ph )
Theorem	ral0 3510	Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
|- A. x e. (/) ph
Theorem	rgenm 3511*	Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
|- ( ( E. x x e. A /\ x e. A ) -> ph )   =>    |- A. x e. A ph
Theorem	ralf0 3512*	The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
|- -. ph   =>    |- ( A. x e. A ph <-> A = (/) )
Theorem	ralm 3513	Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- ( ( E. x x e. A -> A. x e. A ph ) <-> A. x e. A ph )
Theorem	raaanlem 3514*	Special case of raaan 3515 where A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( E. x x e. A -> ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) ) )
Theorem	raaan 3515*	Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
|- F/ y ph   &    |- F/ x ps   =>    |- ( A. x e. A A. y e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. y e. A ps ) )
Theorem	raaanv 3516*	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 3517*	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 3518	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 3519	The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.)
|- ( ph -> DECID k e. A )   &    |- ( ph -> DECID k e. B )   =>    |- ( ph -> DECID k e. ( A u. B ) )
2.1.15  Conditional operator
Syntax	cif 3520	Extend class notation to include the conditional operator. See df-if 3521 for a description. (In older databases this was denoted "ded".)
class if ( ph , A , B )
Definition	df-if 3521*	Define the conditional operator. Read if ( ph , A , B ) as "if ph then A else B". See iftrue 3525 and iffalse 3528 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 825). (Contributed by NM, 15-May-1999.)
|- if ( ph , A , B ) = { x | ( ( x e. A /\ ph ) \/ ( x e. B /\ -. ph ) ) }
Theorem	dfif6 3522*	An alternate definition of the conditional operator df-if 3521 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 3523	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 3524	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 3525	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 3526	Inference associated with iftrue 3525. (Contributed by BJ, 7-Oct-2018.)
|- ph   =>    |- if ( ph , A , B ) = A
Theorem	iftrued 3527	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 3528	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 3529	Inference associated with iffalse 3528. (Contributed by BJ, 7-Oct-2018.)
|- -. ph   =>    |- if ( ph , A , B ) = B
Theorem	iffalsed 3530	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 3531	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 3528 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
|- ( A =/= B -> if ( A = B , C , D ) = D )
Theorem	ifsbdc 3532	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 3533*	Alternate definition of the conditional operator df-if 3521. 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 3534	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 3535	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 3536	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 3537	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 3538	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 3539	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 3540	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
|- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )
Theorem	ifbid 3541	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 3542	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 3543	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 3544	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 3545	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 3546	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 3547	Deduction version of nfif 3548. (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 3548	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 3549	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 3550	Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)
|- ( ( ph /\ ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifbothdadc 3551	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 3552	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 3553	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
|- (DECID ph -> if ( ph , A , A ) = A )
Theorem	eqifdc 3554	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 3555	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 3556	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	ifandc 3557	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	ifmdc 3558	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 )
2.1.16  Power classes
Syntax	cpw 3559	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 3560*	Soundness justification theorem for df-pw 3561. (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 3561*	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 3562	Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ~P A = ~P B )
Theorem	pweqi 3563	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
|- A = B   =>    |- ~P A = ~P B
Theorem	pweqd 3564	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ~P A = ~P B )
Theorem	elpw 3565	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 3566*	Setvar variable membership in a power class (common case). See elpw 3565. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( x e. ~P A <-> x C_ A )
Theorem	elpwg 3567	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 3568	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
|- ( A e. ~P B -> A C_ B )
Theorem	elpwb 3569	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 3570	An element of a power class is a subclass. Deduction form of elpwi 3568. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ~P B )   =>    |- ( ph -> A C_ B )
Theorem	elelpwi 3571	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 3572	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 3573	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 3574	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 3575*	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 3576	Extend class notation to include singleton.
class { A }
Syntax	cpr 3577	Extend class notation to include unordered pair.
class { A , B }
Syntax	ctp 3578	Extend class notation to include unordered triplet.
class { A , B , C }
Syntax	cop 3579	Extend class notation to include ordered pair.
class <. A , B >.
Syntax	cotp 3580	Extend class notation to include ordered triple.
class <. A , B , C >.
Theorem	snjust 3581*	Soundness justification theorem for df-sn 3582. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x = A } = { y | y = A }
Definition	df-sn 3582*	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 3590. (Contributed by NM, 5-Aug-1993.)
|- { A } = { x | x = A }
Definition	df-pr 3583	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so { A , B } = { B , A } as proven by prcom 3652. For a more traditional definition, but requiring a dummy variable, see dfpr2 3595. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = ( { A } u. { B } )
Definition	df-tp 3584	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 3585*	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 3780 and opprc2 3781). For Kuratowski's actual definition when the arguments are sets, see dfop 3757. 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 3585 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3585 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 3586	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 3587	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 3588	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 3589	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 3590	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsng 3591	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 3592	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 3593	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 3594	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	dfpr2 3595*	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 3596	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 3597	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 3598	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 3599	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 3600	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 }