P. 36 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sbcgf 3501	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- F/ x ph   =>    |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
Theorem	sbc19.21g 3502	Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
|- F/ x ph   =>    |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( ph -> [. A / x ]. ps ) ) )
Theorem	sbcg 3503*	Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3501. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> ( [. A / x ]. ph <-> ph ) )
Theorem	sbc2iegf 3504*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
|- F/ x ps   &    |- F/ y ps   &    |- F/ x B e. W   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
Theorem	sbc2ie 3505*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. [. B / y ]. ph <-> ps )
Theorem	sbc2iedv 3506*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
|- A e. _V   &    |- B e. _V   &    |- ( ph -> ( ( x = A /\ y = B ) -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( [. A / x ]. [. B / y ]. ps <-> ch ) )
Theorem	sbc3ie 3507*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. [. B / y ]. [. C / z ]. ph <-> ps )
Theorem	sbccomlem 3508*	Lemma for sbccom 3509. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
|- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )
Theorem	sbccom 3509*	Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
|- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. [. A / x ]. ph )
Theorem	sbcralt 3510*	Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
|- ( ( A e. V /\ F/_ y A ) -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
Theorem	sbcrext 3511*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
|- ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
Theorem	sbcrextOLD 3512*	Obsolete proof of sbcrext 3511 as of 7-Jul-2021. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
Theorem	sbcralg 3513*	Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
Theorem	sbcrex 3514*	Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
|- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph )
Theorem	sbcreu 3515*	Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)
|- ( [. A / x ]. E! y e. B ph <-> E! y e. B [. A / x ]. ph )
Theorem	reu8nf 3516*	Restricted uniqueness using implicit substitution. This version of reu8 3402 uses a non-freeness hypothesis for x and ps instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.)
|- F/ x ps   &    |- F/ x ch   &    |- ( x = w -> ( ph <-> ch ) )   &    |- ( w = y -> ( ch <-> ps ) )   =>    |- ( E! x e. A ph <-> E. x e. A ( ph /\ A. y e. A ( ps -> x = y ) ) )
Theorem	sbcabel 3517*	Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
|- F/_ x B   =>    |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <-> { y | [. A / x ]. ph } e. B ) )
Theorem	rspsbc 3518*	Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2353 and spsbc 3448. See also rspsbca 3519 and rspcsbela 4006. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( A e. B -> ( A. x e. B ph -> [. A / x ]. ph ) )
Theorem	rspsbca 3519*	Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
|- ( ( A e. B /\ A. x e. B ph ) -> [. A / x ]. ph )
Theorem	rspesbca 3520*	Existence form of rspsbca 3519. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( ( A e. B /\ [. A / x ]. ph ) -> E. x e. B ph )
Theorem	spesbc 3521	Existence form of spsbc 3448. (Contributed by Mario Carneiro, 18-Nov-2016.)
|- ( [. A / x ]. ph -> E. x ph )
Theorem	spesbcd 3522	form of spsbc 3448. (Contributed by Mario Carneiro, 9-Feb-2017.)
|- ( ph -> [. A / x ]. ps )   =>    |- ( ph -> E. x ps )
Theorem	sbcth2 3523*	A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
|- ( x e. B -> ph )   =>    |- ( A e. B -> [. A / x ]. ph )
Theorem	ra4v 3524*	Version of ra4 3525 with a dv condition, requiring fewer axioms. This is stdpc5v 1867 for a restricted domain. (Contributed by BJ, 27-Mar-2020.)
|- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )
Theorem	ra4 3525	Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2076 of standard predicate calculus for a restricted domain. See ra4v 3524 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.)
|- F/ x ph   =>    |- ( A. x e. A ( ph -> ps ) -> ( ph -> A. x e. A ps ) )
Theorem	rmo2 3526*	Alternate definition of restricted "at most one." Note that E* x e. A ph is not equivalent to E. y e. A A. x e. A ( ph -> x = y ) (in analogy to reu6 3395); to see this, let A be the empty set. However, one direction of this pattern holds; see rmo2i 3527. (Contributed by NM, 17-Jun-2017.)
|- F/ y ph   =>    |- ( E* x e. A ph <-> E. y A. x e. A ( ph -> x = y ) )
Theorem	rmo2i 3527*	Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
|- F/ y ph   =>    |- ( E. y e. A A. x e. A ( ph -> x = y ) -> E* x e. A ph )
Theorem	rmo3 3528*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
|- F/ y ph   =>    |- ( E* x e. A ph <-> A. x e. A A. y e. A ( ( ph /\ [ y / x ] ph ) -> x = y ) )
Theorem	rmob 3529*	Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
|- ( x = B -> ( ph <-> ps ) )   &    |- ( x = C -> ( ph <-> ch ) )   =>    |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
Theorem	rmoi 3530*	Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
|- ( x = B -> ( ph <-> ps ) )   &    |- ( x = C -> ( ph <-> ch ) )   =>    |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )
Theorem	rmob2 3531*	Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
|- ( x = B -> ( ps <-> ch ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> E* x e. A ps )   &    |- ( ph -> x e. A )   &    |- ( ph -> ps )   =>    |- ( ph -> ( x = B <-> ch ) )
Theorem	rmoi2 3532*	Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
|- ( x = B -> ( ps <-> ch ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> E* x e. A ps )   &    |- ( ph -> x e. A )   &    |- ( ph -> ps )   &    |- ( ph -> ch )   =>    |- ( ph -> x = B )
2.1.10  Proper substitution of classes for sets into classes
Syntax	csb 3533	Extend class notation to include the proper substitution of a class for a set into another class.
class [_ A / x ]_ B
Definition	df-csb 3534*	Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3435, to prevent ambiguity. Theorem sbcel1g 3987 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsb 4004 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
|- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
Theorem	csb2 3535*	Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that x can be free in B but cannot occur in A. (Contributed by NM, 2-Dec-2013.)
|- [_ A / x ]_ B = { y | E. x ( x = A /\ y e. B ) }
Theorem	csbeq1 3536	Analogue of dfsbcq 3437 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- ( A = B -> [_ A / x ]_ C = [_ B / x ]_ C )
Theorem	csbeq2 3537	Substituting into equivalent classes gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
|- ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C )
Theorem	cbvcsb 3538	Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/_ y C   &    |- F/_ x D   &    |- ( x = y -> C = D )   =>    |- [_ A / x ]_ C = [_ A / y ]_ D
Theorem	cbvcsbv 3539*	Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( x = y -> B = C )   =>    |- [_ A / x ]_ B = [_ A / y ]_ C
Theorem	csbeq1d 3540	Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
|- ( ph -> A = B )   =>    |- ( ph -> [_ A / x ]_ C = [_ B / x ]_ C )
Theorem	csbid 3541	Analogue of sbid 2114 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- [_ x / x ]_ A = A
Theorem	csbeq1a 3542	Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- ( x = A -> B = [_ A / x ]_ B )
Theorem	csbco 3543*	Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
Theorem	csbtt 3544	Substitution doesn't affect a constant B (in which x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )
Theorem	csbconstgf 3545	Substitution doesn't affect a constant B (in which x is not free). (Contributed by NM, 10-Nov-2005.)
|- F/_ x B   =>    |- ( A e. V -> [_ A / x ]_ B = B )
Theorem	csbconstg 3546*	Substitution doesn't affect a constant B (in which x does not occur). csbconstgf 3545 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( A e. V -> [_ A / x ]_ B = B )
Theorem	nfcsb1d 3547	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- ( ph -> F/_ x A )   =>    |- ( ph -> F/_ x [_ A / x ]_ B )
Theorem	nfcsb1 3548	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   =>    |- F/_ x [_ A / x ]_ B
Theorem	nfcsb1v 3549*	Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/_ x [_ A / x ]_ B
Theorem	nfcsbd 3550	Deduction version of nfcsb 3551. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x [_ A / y ]_ B )
Theorem	nfcsb 3551	Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x [_ A / y ]_ B
Theorem	csbhypf 3552*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3253 for class substitution version. (Contributed by NM, 19-Dec-2008.)
|- F/_ x A   &    |- F/_ x C   &    |- ( x = A -> B = C )   =>    |- ( y = A -> [_ y / x ]_ B = C )
Theorem	csbiebt 3553*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3557.) (Contributed by NM, 11-Nov-2005.)
|- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
Theorem	csbiedf 3554*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
|- F/ x ph   &    |- ( ph -> F/_ x C )   &    |- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = C )
Theorem	csbieb 3555*	Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
|- A e. _V   &    |- F/_ x C   =>    |- ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C )
Theorem	csbiebg 3556*	Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
|- F/_ x C   =>    |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
Theorem	csbiegf 3557*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( A e. V -> F/_ x C )   &    |- ( x = A -> B = C )   =>    |- ( A e. V -> [_ A / x ]_ B = C )
Theorem	csbief 3558*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- A e. _V   &    |- F/_ x C   &    |- ( x = A -> B = C )   =>    |- [_ A / x ]_ B = C
Theorem	csbie 3559*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)
|- A e. _V   &    |- ( x = A -> B = C )   =>    |- [_ A / x ]_ B = C
Theorem	csbied 3560*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x = A ) -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = C )
Theorem	csbied2 3561*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> A e. V )   &    |- ( ph -> A = B )   &    |- ( ( ph /\ x = B ) -> C = D )   =>    |- ( ph -> [_ A / x ]_ C = D )
Theorem	csbie2t 3562*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3563). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- A e. _V   &    |- B e. _V   =>    |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )
Theorem	csbie2 3563*	Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x = A /\ y = B ) -> C = D )   =>    |- [_ A / x ]_ [_ B / y ]_ C = D
Theorem	csbie2g 3564*	Conversion of implicit substitution to explicit class substitution. This version of csbie 3559 avoids a disjointness condition on x , A and x , D by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
|- ( x = y -> B = C )   &    |- ( y = A -> C = D )   =>    |- ( A e. V -> [_ A / x ]_ B = D )
Theorem	cbvralcsf 3565	A more general version of cbvralf 3165 that doesn't require A and B to be distinct from x or y. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. B ps )
Theorem	cbvrexcsf 3566	A more general version of cbvrexf 3166 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. B ps )
Theorem	cbvreucsf 3567	A more general version of cbvreuv 3173 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. B ps )
Theorem	cbvrabcsf 3568	A more general version of cbvrab 3198 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)
|- F/_ y A   &    |- F/_ x B   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> A = B )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x e. A | ph } = { y e. B | ps }
Theorem	cbvralv2 3569*	Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = y -> A = B )   =>    |- ( A. x e. A ps <-> A. y e. B ch )
Theorem	cbvrexv2 3570*	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
|- ( x = y -> ( ps <-> ch ) )   &    |- ( x = y -> A = B )   =>    |- ( E. x e. A ps <-> E. y e. B ch )
2.1.11  Define basic set operations and relations
Syntax	cdif 3571	Extend class notation to include class difference (read: " A minus B").
class ( A \ B )
Syntax	cun 3572	Extend class notation to include union of two classes (read: " A union B").
class ( A u. B )
Syntax	cin 3573	Extend class notation to include the intersection of two classes (read: " A intersect B").
class ( A i^i B )
Syntax	wss 3574	Extend wff notation to include the subclass relation. This is read " A is a subclass of B " or " B includes A." When A exists as a set, it is also read " A is a subset of B."
wff A C_ B
Syntax	wpss 3575	Extend wff notation with proper subclass relation.
wff A C. B
Theorem	difjust 3576*	Soundness justification theorem for df-dif 3577. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A /\ -. x e. B ) } = { y | ( y e. A /\ -. y e. B ) }
Definition	df-dif 3577*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ( { 1 , 3 } \ { 1 , 8 } ) = { 3 } (ex-dif 27280). Contrast this operation with union ( A u. B ) (df-un 3579) and intersection ( A i^i B ) (df-in 3581). Several notations are used in the literature; we chose the \ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " A excludes B " to mean A \ B. We will use " B is removed from A " to mean A \ { B } i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)
|- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
Theorem	unjust 3578*	Soundness justification theorem for df-un 3579. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A \/ x e. B ) } = { y | ( y e. A \/ y e. B ) }
Definition	df-un 3579*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ( { 1 , 3 } u. { 1 , 8 } ) = { 1 , 3 , 8 } (ex-un 27281). Contrast this operation with difference ( A \ B ) (df-dif 3577) and intersection ( A i^i B ) (df-in 3581). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3859. For union defined in terms of intersection, see dfun3 3865. (Contributed by NM, 23-Aug-1993.)
|- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
Theorem	injust 3580*	Soundness justification theorem for df-in 3581. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- { x | ( x e. A /\ x e. B ) } = { y | ( y e. A /\ y e. B ) }
Definition	df-in 3581*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ( { 1 , 3 } i^i { 1 , 8 } ) = { 1 } (ex-in 27282). Contrast this operation with union ( A u. B ) (df-un 3579) and difference ( A \ B ) (df-dif 3577). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3860 and dfin4 3867. For intersection defined in terms of union, see dfin3 3866. (Contributed by NM, 29-Apr-1994.)
|- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
Theorem	dfin5 3582*	Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
|- ( A i^i B ) = { x e. A | x e. B }
Theorem	dfdif2 3583*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
|- ( A \ B ) = { x e. A | -. x e. B }
Theorem	eldif 3584	Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
|- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
Theorem	eldifd 3585	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3584. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ph -> -. A e. C )   =>    |- ( ph -> A e. ( B \ C ) )
Theorem	eldifad 3586	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3584. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ( B \ C ) )   =>    |- ( ph -> A e. B )
Theorem	eldifbd 3587	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3584. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ( B \ C ) )   =>    |- ( ph -> -. A e. C )
2.1.12  Subclasses and subsets
Definition	df-ss 3588	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, { 1 , 2 } C_ { 1 , 2 , 3 } (ex-ss 27284). Note that A C_ A (proved in ssid 3624). Contrast this relationship with the relationship A C. B (as will be defined in df-pss 3590). For a more traditional definition, but requiring a dummy variable, see dfss2 3591. Other possible definitions are given by dfss3 3592, dfss4 3858, sspss 3706, ssequn1 3783, ssequn2 3786, sseqin2 3817, and ssdif0 3942. (Contributed by NM, 27-Apr-1994.)
|- ( A C_ B <-> ( A i^i B ) = A )
Theorem	dfss 3589	Variant of subclass definition df-ss 3588. (Contributed by NM, 21-Jun-1993.)
|- ( A C_ B <-> A = ( A i^i B ) )
Definition	df-pss 3590	Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, { 1 , 2 } C. { 1 , 2 , 3 } (ex-pss 27285). Note that -. A C. A (proved in pssirr 3707). Contrast this relationship with the relationship A C_ B (as defined in df-ss 3588). Other possible definitions are given by dfpss2 3692 and dfpss3 3693. (Contributed by NM, 7-Feb-1996.)
|- ( A C. B <-> ( A C_ B /\ A =/= B ) )
Theorem	dfss2 3591*	Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
|- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
Theorem	dfss3 3592*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B <-> A. x e. A x e. B )
Theorem	dfss6 3593*	Alternate definition of subclass relationship. (Contributed by RP, 16-Apr-2020.)
|- ( A C_ B <-> -. E. x ( x e. A /\ -. x e. B ) )
Theorem	dfss2f 3594	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
Theorem	dfss3f 3595	Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B <-> A. x e. A x e. B )
Theorem	nfss 3596	If x is not free in A and B, it is not free in A C_ B. (Contributed by NM, 27-Dec-1996.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/ x A C_ B
Theorem	ssel 3597	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- ( A C_ B -> ( C e. A -> C e. B ) )
Theorem	ssel2 3598	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
|- ( ( A C_ B /\ C e. A ) -> C e. B )
Theorem	sseli 3599	Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- A C_ B   =>    |- ( C e. A -> C e. B )
Theorem	sselii 3600	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
|- A C_ B   &    |- C e. A   =>    |- C e. B