P. 32 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbvcsbw 3101*	Version of cbvcsb 3102 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
|- F/_ y C   &    |- F/_ x D   &    |- ( x = y -> C = D )   =>    |- [_ A / x ]_ C = [_ A / y ]_ D
Theorem	cbvcsb 3102	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 3103*	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 3104	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 3105	Analog of sbid 1798 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- [_ x / x ]_ A = A
Theorem	csbeq1a 3106	Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
|- ( x = A -> B = [_ A / x ]_ B )
Theorem	csbco 3107*	Composition law for chained substitutions into a class. Use the weaker csbcow 3108 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
Theorem	csbcow 3108*	Composition law for chained substitutions into a class. Version of csbco 3107 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by GG, 25-Aug-2024.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
Theorem	csbtt 3109	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 3110	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 3111*	Substitution doesn't affect a constant B (in which x is not free). csbconstgf 3110 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( A e. V -> [_ A / x ]_ B = B )
Theorem	sbcel12g 3112	Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
Theorem	sbceqg 3113	Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
Theorem	sbcnel12g 3114	Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )
Theorem	sbcne12g 3115	Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
|- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )
Theorem	sbcel1g 3116*	Move proper substitution in and out of a membership relation. Note that the scope of [. A / x ]. is the wff B e. C, whereas the scope of [_ A / x ]_ is the class B. (Contributed by NM, 10-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. C ) )
Theorem	sbceq1g 3117*	Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = C ) )
Theorem	sbcel2g 3118*	Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B e. C <-> B e. [_ A / x ]_ C ) )
Theorem	sbceq2g 3119*	Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. B = C <-> B = [_ A / x ]_ C ) )
Theorem	csbcomg 3120*	Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
|- ( ( A e. V /\ B e. W ) -> [_ A / x ]_ [_ B / y ]_ C = [_ B / y ]_ [_ A / x ]_ C )
Theorem	csbeq2 3121	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	csbeq2d 3122	Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- F/ x ph   &    |- ( ph -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )
Theorem	csbeq2dv 3123*	Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- ( ph -> B = C )   =>    |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )
Theorem	csbeq2i 3124	Formula-building inference for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
|- B = C   =>    |- [_ A / x ]_ B = [_ A / x ]_ C
Theorem	csbvarg 3125	The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
|- ( A e. V -> [_ A / x ]_ x = A )
Theorem	sbccsbg 3126*	Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
|- ( A e. V -> ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } ) )
Theorem	sbccsb2g 3127	Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
|- ( A e. V -> ( [. A / x ]. ph <-> A e. [_ A / x ]_ { x | ph } ) )
Theorem	nfcsb1d 3128	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 3129	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 3130*	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	nfsbcdw 3131*	Version of nfsbcd 3022 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x [. A / y ]. ps )
Theorem	nfsbcw 3132*	Bound-variable hypothesis builder for class substitution. Version of nfsbc 3023 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x [. A / y ]. ph
Theorem	nfcsbd 3133	Deduction version of nfcsb 3135. (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	nfcsbw 3134*	Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3135 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x [_ A / y ]_ B
Theorem	nfcsb 3135	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 3136*	Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2824 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 3137*	Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3141.) (Contributed by NM, 11-Nov-2005.)
|- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
Theorem	csbiedf 3138*	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 3139*	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 3140*	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 3141*	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 3142*	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 3143*	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 3144*	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 3145*	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 3146*	Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3147). (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 3147*	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 3148*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3037 avoids a disjointness condition on x and A 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	sbcnestgf 3149	Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
|- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
Theorem	csbnestgf 3150	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
Theorem	sbcnestg 3151*	Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
Theorem	csbnestg 3152*	Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
Theorem	csbnest1g 3153	Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
|- ( A e. V -> [_ A / x ]_ [_ B / x ]_ C = [_ [_ A / x ]_ B / x ]_ C )
Theorem	csbidmg 3154*	Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)
|- ( A e. V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
Theorem	sbcco3g 3155*	Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )
Theorem	csbco3g 3156*	Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
|- ( x = A -> B = C )   =>    |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )
Theorem	rspcsbela 3157*	Special case related to rspsbc 3085. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
|- ( ( A e. B /\ A. x e. B C e. D ) -> [_ A / x ]_ C e. D )
Theorem	sbnfc2 3158*	Two ways of expressing " x is (effectively) not free in A." (Contributed by Mario Carneiro, 14-Oct-2016.)
|- ( F/_ x A <-> A. y A. z [_ y / x ]_ A = [_ z / x ]_ A )
Theorem	csbabg 3159*	Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( A e. V -> [_ A / x ]_ { y | ph } = { y | [. A / x ]. ph } )
Theorem	cbvralcsf 3160	A more general version of cbvralf 2731 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 3161	A more general version of cbvrexf 2732 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 3162	A more general version of cbvreuv 2741 that has no distinct variable rextrictions. 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 3163	A more general version of cbvrab 2771 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 3164*	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 3165*	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 )
Theorem	rspc2vd 3166*	Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class D for the second set variable y may depend on the first set variable x. (Contributed by AV, 29-Mar-2021.)
|- ( x = A -> ( th <-> ch ) )   &    |- ( y = B -> ( ch <-> ps ) )   &    |- ( ph -> A e. C )   &    |- ( ( ph /\ x = A ) -> D = E )   &    |- ( ph -> B e. E )   =>    |- ( ph -> ( A. x e. C A. y e. D th -> ps ) )
2.1.11  Define basic set operations and relations
Syntax	cdif 3167	Extend class notation to include class difference (read: " A minus B").
class ( A \ B )
Syntax	cun 3168	Extend class notation to include union of two classes (read: " A union B").
class ( A u. B )
Syntax	cin 3169	Extend class notation to include the intersection of two classes (read: " A intersect B").
class ( A i^i B )
Syntax	wss 3170	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
Theorem	difjust 3171*	Soundness justification theorem for df-dif 3172. (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 3172*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union ( A u. B ) (df-un 3174) and intersection ( A i^i B ) (df-in 3176). 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 3173*	Soundness justification theorem for df-un 3174. (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 3174*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference ( A \ B ) (df-dif 3172) and intersection ( A i^i B ) (df-in 3176). (Contributed by NM, 23-Aug-1993.)
|- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
Theorem	injust 3175*	Soundness justification theorem for df-in 3176. (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 3176*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union ( A u. B ) (df-un 3174) and difference ( A \ B ) (df-dif 3172). (Contributed by NM, 29-Apr-1994.)
|- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
Theorem	dfin5 3177*	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 3178*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
|- ( A \ B ) = { x e. A | -. x e. B }
Theorem	eldif 3179	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 3180	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3179. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ph -> -. A e. C )   =>    |- ( ph -> A e. ( B \ C ) )
Theorem	eldifad 3181	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3179. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ( B \ C ) )   =>    |- ( ph -> A e. B )
Theorem	eldifbd 3182	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3179. (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 3183	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that A C_ A (proved in ssid 3217). For a more traditional definition, but requiring a dummy variable, see ssalel 3185. Other possible definitions are given by dfss3 3186, ssequn1 3347, ssequn2 3350, and sseqin2 3396. (Contributed by NM, 27-Apr-1994.)
|- ( A C_ B <-> ( A i^i B ) = A )
Theorem	dfss 3184	Variant of subclass definition df-ss 3183. (Contributed by NM, 3-Sep-2004.)
|- ( A C_ B <-> A = ( A i^i B ) )
Theorem	ssalel 3185*	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 3186*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B <-> A. x e. A x e. B )
Theorem	dfss2 3187	Alternate definition of the subclass relationship between two classes. Exercise 9 of [TakeutiZaring] p. 18. This is another name for df-ss 3183 which is more consistent with the naming in the Metamath Proof Explorer. (Contributed by NM, 27-Apr-1994.)
|- ( A C_ B <-> ( A i^i B ) = A )
Theorem	dfss2f 3188	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 3189	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 3190	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 3191	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- ( A C_ B -> ( C e. A -> C e. B ) )
Theorem	ssel2 3192	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
|- ( ( A C_ B /\ C e. A ) -> C e. B )
Theorem	sseli 3193	Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- A C_ B   =>    |- ( C e. A -> C e. B )
Theorem	sselii 3194	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
|- A C_ B   &    |- C e. A   =>    |- C e. B
Theorem	sselid 3195	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
|- A C_ B   &    |- ( ph -> C e. A )   =>    |- ( ph -> C e. B )
Theorem	sseld 3196	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C e. A -> C e. B ) )
Theorem	sselda 3197	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
|- ( ph -> A C_ B )   =>    |- ( ( ph /\ C e. A ) -> C e. B )
Theorem	sseldd 3198	Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> C e. A )   =>    |- ( ph -> C e. B )
Theorem	ssneld 3199	If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( -. C e. B -> -. C e. A ) )
Theorem	ssneldd 3200	If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   &    |- ( ph -> -. C e. B )   =>    |- ( ph -> -. C e. A )