P. 42 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfif4 4101*	Alternate definition of the conditional operator df-if 4087. Note that ph is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
|- C = { x | ph }   =>    |- if ( ph , A , B ) = ( ( A u. B ) i^i ( ( A u. ( _V \ C ) ) i^i ( B u. C ) ) )
Theorem	dfif5 4102*	Alternate definition of the conditional operator df-if 4087. Note that ph is independent of x i.e. a constant true or false (see also ab0orv 3953). (Contributed by Gérard Lang, 18-Aug-2013.)
|- C = { x | ph }   =>    |- if ( ph , A , B ) = ( ( A i^i B ) u. ( ( ( A \ B ) i^i C ) u. ( ( B \ A ) i^i ( _V \ C ) ) ) )
Theorem	ifeq12 4103	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 4104	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 4105	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 4106	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 4107	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
|- ( ( ph <-> ps ) -> if ( ph , A , B ) = if ( ps , A , B ) )
Theorem	ifbid 4108	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 4109	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 4110	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 4111	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 4112	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 4113	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 4114	Deduction version of nfif 4115. (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 4115	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	ifeq1da 4116	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ ps ) -> A = B )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifeq2da 4117	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ -. ps ) -> A = B )   =>    |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )
Theorem	ifeq12da 4118	Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
|- ( ( ph /\ ps ) -> A = C )   &    |- ( ( ph /\ -. ps ) -> B = D )   =>    |- ( ph -> if ( ps , A , B ) = if ( ps , C , D ) )
Theorem	ifbieq12d2 4119	Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ( ph /\ ps ) -> A = C )   &    |- ( ( ph /\ -. ps ) -> B = D )   =>    |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )
Theorem	ifclda 4120	Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ ps ) -> A e. C )   &    |- ( ( ph /\ -. ps ) -> B e. C )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifeqda 4121	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 -> if ( ps , A , B ) = C )
Theorem	elimif 4122	Elimination of a conditional operator contained in a wff ps. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
|- ( if ( ph , A , B ) = A -> ( ps <-> ch ) )   &    |- ( if ( ph , A , B ) = B -> ( ps <-> th ) )   =>    |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )
Theorem	ifbothda 4123	A wff th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   &    |- ( ( et /\ ph ) -> ps )   &    |- ( ( et /\ -. ph ) -> ch )   =>    |- ( et -> th )
Theorem	ifboth 4124	A wff th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   =>    |- ( ( ps /\ ch ) -> th )
Theorem	ifid 4125	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
|- if ( ph , A , A ) = A
Theorem	eqif 4126	Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
|- ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) )
Theorem	ifval 4127	Another expression of the value of the if predicate, analogous to eqif 4126. See also the more specialized iftrue 4092 and iffalse 4095. (Contributed by BJ, 6-Apr-2019.)
|- ( A = if ( ph , B , C ) <-> ( ( ph -> A = B ) /\ ( -. ph -> A = C ) ) )
Theorem	elif 4128	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	ifel 4129	Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
|- ( if ( ph , A , B ) e. C <-> ( ( ph /\ A e. C ) \/ ( -. ph /\ B e. C ) ) )
Theorem	ifcl 4130	Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
|- ( ( A e. C /\ B e. C ) -> if ( ph , A , B ) e. C )
Theorem	ifcld 4131	Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifeqor 4132	The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( if ( ph , A , B ) = A \/ if ( ph , A , B ) = B )
Theorem	ifnot 4133	Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
|- if ( -. ph , A , B ) = if ( ph , B , A )
Theorem	ifan 4134	Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B )
Theorem	ifor 4135	Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) )
Theorem	2if2 4136	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 -> D = if ( ps , A , if ( th , B , C ) ) )
Theorem	ifcomnan 4137	Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)
|- ( -. ( ph /\ ps ) -> if ( ph , A , if ( ps , B , C ) ) = if ( ps , B , if ( ph , A , C ) ) )
Theorem	csbif 4138	Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
|- [_ A / x ]_ if ( ph , B , C ) = if ( [. A / x ]. ph , [_ A / x ]_ B , [_ A / x ]_ C )
Theorem	dedth 4139	Weak deduction theorem that eliminates a hypothesis ph, making it become an antecedent. We assume that a proof exists for ph when the class variable A is replaced with a specific class B. The hypothesis ch should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 4146. If the inference has other hypotheses with class variable A, these can be kept by assigning keephyp 4152 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 15-May-1999.)
|- ( A = if ( ph , A , B ) -> ( ps <-> ch ) )   &    |- ch   =>    |- ( ph -> ps )
Theorem	dedth2h 4140	Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 4143 but requires that each hypothesis has exactly one class variable. See also comments in dedth 4139. (Contributed by NM, 15-May-1999.)
|- ( A = if ( ph , A , C ) -> ( ch <-> th ) )   &    |- ( B = if ( ps , B , D ) -> ( th <-> ta ) )   &    |- ta   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	dedth3h 4141	Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4140. (Contributed by NM, 15-May-1999.)
|- ( A = if ( ph , A , D ) -> ( th <-> ta ) )   &    |- ( B = if ( ps , B , R ) -> ( ta <-> et ) )   &    |- ( C = if ( ch , C , S ) -> ( et <-> ze ) )   &    |- ze   =>    |- ( ( ph /\ ps /\ ch ) -> th )
Theorem	dedth4h 4142	Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4140. (Contributed by NM, 16-May-1999.)
|- ( A = if ( ph , A , R ) -> ( ta <-> et ) )   &    |- ( B = if ( ps , B , S ) -> ( et <-> ze ) )   &    |- ( C = if ( ch , C , F ) -> ( ze <-> si ) )   &    |- ( D = if ( th , D , G ) -> ( si <-> rh ) )   &    |- rh   =>    |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) -> ta )
Theorem	dedth2v 4143	Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4140 is simpler to use. See also comments in dedth 4139. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
|- ( A = if ( ph , A , C ) -> ( ps <-> ch ) )   &    |- ( B = if ( ph , B , D ) -> ( ch <-> th ) )   &    |- th   =>    |- ( ph -> ps )
Theorem	dedth3v 4144	Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4143. (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 4145	Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4143. (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 4146	Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 4139. (Contributed by NM, 15-May-1999.)
|- ( A = if ( ph , A , B ) -> ( ph <-> ps ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> ps ) )   &    |- ch   =>    |- ps
Theorem	elimhyp2v 4147	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 4148	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 4149	Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 4139). (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 4150	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 4151	Version of elimhyp 4146 where the hypothesis is deduced from the final antecedent. See divalg 15126 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 4152	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 4153	Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4139). (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 4154	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 4155	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 4156	Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
|- A e. _V   &    |- B e. _V   =>    |- if ( ph , A , B ) e. _V
Theorem	ifexg 4157	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 4158	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 4159*	Soundness justification theorem for df-pw 4160. (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 4160*	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 27286). We will later introduce the Axiom of Power Sets ax-pow 4843, which can be expressed in class notation per pwexg 4850. Still later we will prove, in hashpw 13223, 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 4161	Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)
|- ( A = B -> ~P A = ~P B )
Theorem	pweqi 4162	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
|- A = B   =>    |- ~P A = ~P B
Theorem	pweqd 4163	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ~P A = ~P B )
Theorem	elpw 4164	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 4165*	Setvar variable membership in a power class (common case). See elpw 4164. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( x e. ~P A <-> x C_ A )
Theorem	elpwg 4166	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4827. (Contributed by NM, 6-Aug-2000.)
|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )
Theorem	elpwd 4167	Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> A e. ~P B )
Theorem	elpwi 4168	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
|- ( A e. ~P B -> A C_ B )
Theorem	elpwb 4169	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 4170	An element of a power class is a subclass. Deduction form of elpwi 4168. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ~P B )   =>    |- ( ph -> A C_ B )
Theorem	elelpwi 4171	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 4172	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 4173	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 4174	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 4175*	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 4176*	Soundness justification theorem for df-sn 4178. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x = A } = { y | y = A }
Syntax	csn 4177	Extend class notation to include singleton.
class { A }
Definition	df-sn 4178*	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 4190. (Contributed by NM, 21-Jun-1993.)
|- { A } = { x | x = A }
Syntax	cpr 4179	Extend class notation to include unordered pair.
class { A , B }
Definition	df-pr 4180	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For example, A e. { 1 , -u 1 } -> ( A ^ 2 ) = 1 (ex-pr 27287). They are unordered, so { A , B } = { B , A } as proven by prcom 4267. For a more traditional definition, but requiring a dummy variable, see dfpr2 4195. { A , A } is also an unordered pair, but also a singleton because of { A } = { A , A } (see dfsn2 4190). Therefore, { A , B } is called a proper (unordered) pair iff A =/= B and A and B are sets. (Contributed by NM, 21-Jun-1993.)
|- { A , B } = ( { A } u. { B } )
Syntax	ctp 4181	Extend class notation to include unordered triplet.
class { A , B , C }
Definition	df-tp 4182	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 4183	Extend class notation to include ordered pair.
class <. A , B >.
Definition	df-op 4184*	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 4425, opprc2 4426, and 0nelop 4960). For Kuratowski's actual definition when the arguments are sets, see dfop 4401. For the justifying theorem (for sets) see opth 4945. See dfopif 4399 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 4184 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 4184 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 4976. 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 8515, 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 5167. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 13057. An ordered pair of real numbers can also be represented by a complex number as shown by cru 11012. 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 alternate 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 4399. (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 4185	Extend class notation to include ordered triple.
class <. A , B , C >.
Definition	df-ot 4186	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 4187	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 4188	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 4189	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 4190	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsng 4191	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 4192	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 4193	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 4194	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	dfpr2 4195*	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 4196	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	elpri 4197	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	elpr 4198	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 4199	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.) (Proof shortened by JJ, 23-Jul-2021.)
|- B e. _V   &    |- C e. _V   =>    |- ( A e. { B , C } <-> ( A = B \/ A = C ) )
Theorem	elpr2OLD 4200	Obsolete proof of elpr2 4199 as of 23-Jul-2021. (Contributed by NM, 14-Oct-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
|- B e. _V   &    |- C e. _V   =>    |- ( A e. { B , C } <-> ( A = B \/ A = C ) )