P. 41 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pwpwssunieq 4001*	The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
|- { x | U. x = A } C_ ~P ~P A
Theorem	elpwuni 4002	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
|- ( B e. A -> ( A C_ ~P B <-> U. A = B ) )
Theorem	iinpw 4003*	The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
|- ~P |^| A = |^|_ x e. A ~P x
Theorem	iunpwss 4004*	Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
|- U_ x e. A ~P x C_ ~P U. A
Theorem	rintm 4005*	Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- ( ( X C_ ~P A /\ E. x x e. X ) -> ( A i^i |^| X ) = |^| X )
2.1.21  Disjointness
Syntax	wdisj 4006	Extend wff notation to include the statement that a family of classes B ( x ), for x e. A, is a disjoint family.
wff Disj x e. A B
Definition	df-disj 4007*	A collection of classes B ( x ) is disjoint when for each element y, it is in B ( x ) for at most one x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
|- (Disj x e. A B <-> A. y E* x e. A y e. B )
Theorem	dfdisj2 4008*	Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
|- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )
Theorem	disjss2 4009	If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A. x e. A B C_ C -> (Disj x e. A C -> Disj x e. A B ) )
Theorem	disjeq2 4010	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A. x e. A B = C -> (Disj x e. A B <-> Disj x e. A C ) )
Theorem	disjeq2dv 4011*	Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> (Disj x e. A B <-> Disj x e. A C ) )
Theorem	disjss1 4012*	A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )
Theorem	disjeq1 4013*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )
Theorem	disjeq1d 4014*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ph -> A = B )   =>    |- ( ph -> (Disj x e. A C <-> Disj x e. B C ) )
Theorem	disjeq12d 4015*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> (Disj x e. A C <-> Disj x e. B D ) )
Theorem	cbvdisj 4016*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/_ y B   &    |- F/_ x C   &    |- ( x = y -> B = C )   =>    |- (Disj x e. A B <-> Disj y e. A C )
Theorem	cbvdisjv 4017*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
|- ( x = y -> B = C )   =>    |- (Disj x e. A B <-> Disj y e. A C )
Theorem	nfdisjv 4018*	Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/ yDisj x e. A B
Theorem	nfdisj1 4019	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- F/ xDisj x e. A B
Theorem	disjnim 4020*	If a collection B ( i ) for i e. A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
|- ( i = j -> B = C )   =>    |- (Disj i e. A B -> A. i e. A A. j e. A ( i =/= j -> ( B i^i C ) = (/) ) )
Theorem	disjnims 4021*	If a collection B ( i ) for i e. A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
|- (Disj x e. A B -> A. i e. A A. j e. A ( i =/= j -> ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) )
Theorem	disji2 4022*	Property of a disjoint collection: if B ( X ) = C and B ( Y ) = D, and X =/= Y, then C and D are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- ( x = X -> B = C )   &    |- ( x = Y -> B = D )   =>    |- ( (Disj x e. A B /\ ( X e. A /\ Y e. A ) /\ X =/= Y ) -> ( C i^i D ) = (/) )
Theorem	invdisj 4023*	If there is a function C ( y ) such that C ( y ) = x for all y e. B ( x ), then the sets B ( x ) for distinct x e. A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
|- ( A. x e. A A. y e. B C = x -> Disj x e. A B )
Theorem	disjiun 4024*	A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
|- ( (Disj x e. A B /\ ( C C_ A /\ D C_ A /\ ( C i^i D ) = (/) ) ) -> ( U_ x e. C B i^i U_ x e. D B ) = (/) )
Theorem	sndisj 4025	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A { x }
Theorem	0disj 4026	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. A (/)
Theorem	disjxsn 4027*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. { A } B
Theorem	disjx0 4028	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
|- Disj x e. (/) B
2.1.22  Binary relations
Syntax	wbr 4029	Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff A R B
Definition	df-br 4030	Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when R is a proper class (see for example iprc 4930). (Contributed by NM, 31-Dec-1993.)
|- ( A R B <-> <. A , B >. e. R )
Theorem	breq 4031	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
|- ( R = S -> ( A R B <-> A S B ) )
Theorem	breq1 4032	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( A R C <-> B R C ) )
Theorem	breq2 4033	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
|- ( A = B -> ( C R A <-> C R B ) )
Theorem	breq12 4034	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
Theorem	breqi 4035	Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
|- R = S   =>    |- ( A R B <-> A S B )
Theorem	breq1i 4036	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( A R C <-> B R C )
Theorem	breq2i 4037	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- A = B   =>    |- ( C R A <-> C R B )
Theorem	breq12i 4038	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|- A = B   &    |- C = D   =>    |- ( A R C <-> B R D )
Theorem	breq1d 4039	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A R C <-> B R C ) )
Theorem	breqd 4040	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C A D <-> C B D ) )
Theorem	breq2d 4041	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C R A <-> C R B ) )
Theorem	breq12d 4042	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A R C <-> B R D ) )
Theorem	breq123d 4043	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
|- ( ph -> A = B )   &    |- ( ph -> R = S )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A R C <-> B S D ) )
Theorem	breqdi 4044	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
|- ( ph -> A = B )   &    |- ( ph -> C A D )   =>    |- ( ph -> C B D )
Theorem	breqan12d 4045	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )
Theorem	breqan12rd 4046	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A R C <-> B R D ) )
Theorem	eqnbrtrd 4047	Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
|- ( ph -> A = B )   &    |- ( ph -> -. B R C )   =>    |- ( ph -> -. A R C )
Theorem	nbrne1 4048	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
|- ( ( A R B /\ -. A R C ) -> B =/= C )
Theorem	nbrne2 4049	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
|- ( ( A R C /\ -. B R C ) -> A =/= B )
Theorem	eqbrtri 4050	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B R C   =>    |- A R C
Theorem	eqbrtrd 4051	Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	eqbrtrri 4052	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- A R C   =>    |- B R C
Theorem	eqbrtrrd 4053	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A = B )   &    |- ( ph -> A R C )   =>    |- ( ph -> B R C )
Theorem	breqtri 4054	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- B = C   =>    |- A R C
Theorem	breqtrd 4055	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A R C )
Theorem	breqtrri 4056	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
|- A R B   &    |- C = B   =>    |- A R C
Theorem	breqtrrd 4057	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )
Theorem	3brtr3i 4058	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
|- A R B   &    |- A = C   &    |- B = D   =>    |- C R D
Theorem	3brtr4i 4059	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
|- A R B   &    |- C = A   &    |- D = B   =>    |- C R D
Theorem	3brtr3d 4060	Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
|- ( ph -> A R B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C R D )
Theorem	3brtr4d 4061	Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
|- ( ph -> A R B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C R D )
Theorem	3brtr3g 4062	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
|- ( ph -> A R B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C R D )
Theorem	3brtr4g 4063	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
|- ( ph -> A R B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C R D )
Theorem	eqbrtrid 4064	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- A = B   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	eqbrtrrid 4065	B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
|- B = A   &    |- ( ph -> B R C )   =>    |- ( ph -> A R C )
Theorem	breqtrid 4066	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- A R B   &    |- ( ph -> B = C )   =>    |- ( ph -> A R C )
Theorem	breqtrrid 4067	B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- A R B   &    |- ( ph -> C = B )   =>    |- ( ph -> A R C )
Theorem	eqbrtrdi 4068	A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
|- ( ph -> A = B )   &    |- B R C   =>    |- ( ph -> A R C )
Theorem	eqbrtrrdi 4069	A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
|- ( ph -> B = A )   &    |- B R C   =>    |- ( ph -> A R C )
Theorem	breqtrdi 4070	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
|- ( ph -> A R B )   &    |- B = C   =>    |- ( ph -> A R C )
Theorem	breqtrrdi 4071	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
|- ( ph -> A R B )   &    |- C = B   =>    |- ( ph -> A R C )
Theorem	ssbrd 4072	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C A D -> C B D ) )
Theorem	ssbri 4073	Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
|- A C_ B   =>    |- ( C A D -> C B D )
Theorem	nfbrd 4074	Deduction version of bound-variable hypothesis builder nfbr 4075. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x R )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/ x A R B )
Theorem	nfbr 4075	Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x A   &    |- F/_ x R   &    |- F/_ x B   =>    |- F/ x A R B
Theorem	brab1 4076*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
|- ( x R A <-> x e. { z | z R A } )
Theorem	br0 4077	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
|- -. A (/) B
Theorem	brne0 4078	If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4079. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
|- ( A R B -> R =/= (/) )
Theorem	brm 4079*	If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
|- ( A R B -> E. x x e. R )
Theorem	brun 4080	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
|- ( A ( R u. S ) B <-> ( A R B \/ A S B ) )
Theorem	brin 4081	The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
|- ( A ( R i^i S ) B <-> ( A R B /\ A S B ) )
Theorem	brdif 4082	The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
|- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )
Theorem	sbcbrg 4083	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B [_ A / x ]_ R [_ A / x ]_ C ) )
Theorem	sbcbr12g 4084*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R [_ A / x ]_ C ) )
Theorem	sbcbr1g 4085*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
|- ( A e. D -> ( [. A / x ]. B R C <-> [_ A / x ]_ B R C ) )
Theorem	sbcbr2g 4086*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
|- ( A e. D -> ( [. A / x ]. B R C <-> B R [_ A / x ]_ C ) )
Theorem	brralrspcev 4087*	Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
|- ( ( B e. X /\ A. y e. Y A R B ) -> E. x e. X A. y e. Y A R x )
Theorem	brimralrspcev 4088*	Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
|- ( ( B e. X /\ A. y e. Y ( ( ph /\ A R B ) -> ps ) ) -> E. x e. X A. y e. Y ( ( ph /\ A R x ) -> ps ) )
2.1.23  Ordered-pair class abstractions (class builders)
Syntax	copab 4089	Extend class notation to include ordered-pair class abstraction (class builder).
class { <. x , y >. | ph }
Syntax	cmpt 4090	Extend the definition of a class to include maps-to notation for defining a function via a rule.
class ( x e. A |-> B )
Definition	df-opab 4091*	Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually x and y are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
|- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
Definition	df-mpt 4092*	Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from x (in A) to B ( x )". The class expression B is the value of the function at x and normally contains the variable x. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
Theorem	opabss 4093*	The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { <. x , y >. | x R y } C_ R
Theorem	opabbid 4094	Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )
Theorem	opabbidv 4095*	Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> { <. x , y >. | ps } = { <. x , y >. | ch } )
Theorem	opabbii 4096	Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
|- ( ph <-> ps )   =>    |- { <. x , y >. | ph } = { <. x , y >. | ps }
Theorem	nfopab 4097*	Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
|- F/ z ph   =>    |- F/_ z { <. x , y >. | ph }
Theorem	nfopab1 4098	The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x { <. x , y >. | ph }
Theorem	nfopab2 4099	The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ y { <. x , y >. | ph }
Theorem	cbvopab 4100*	Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
|- F/ z ph   &    |- F/ w ph   &    |- F/ x ps   &    |- F/ y ps   &    |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- { <. x , y >. | ph } = { <. z , w >. | ps }