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	3sstr3i 3101	Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A C_ B   &    |- A = C   &    |- B = D   =>    |- C C_ D
Theorem	3sstr4i 3102	Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A C_ B   &    |- C = A   &    |- D = B   =>    |- C C_ D
Theorem	3sstr3g 3103	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
|- ( ph -> A C_ B )   &    |- A = C   &    |- B = D   =>    |- ( ph -> C C_ D )
Theorem	3sstr4g 3104	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- ( ph -> A C_ B )   &    |- C = A   &    |- D = B   =>    |- ( ph -> C C_ D )
Theorem	3sstr3d 3105	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
|- ( ph -> A C_ B )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> C C_ D )
Theorem	3sstr4d 3106	Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- ( ph -> A C_ B )   &    |- ( ph -> C = A )   &    |- ( ph -> D = B )   =>    |- ( ph -> C C_ D )
Theorem	syl5eqss 3107	B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- A = B   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	syl5eqssr 3108	B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- B = A   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	syl6sseq 3109	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- B = C   =>    |- ( ph -> A C_ C )
Theorem	syl6sseqr 3110	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- C = B   =>    |- ( ph -> A C_ C )
Theorem	sseqtrid 3111	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- B C_ A   &    |- ( ph -> A = C )   =>    |- ( ph -> B C_ C )
Theorem	sseqtrrid 3112	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- B C_ A   &    |- ( ph -> C = A )   =>    |- ( ph -> B C_ C )
Theorem	syl6eqss 3113	A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> A = B )   &    |- B C_ C   =>    |- ( ph -> A C_ C )
Theorem	syl6eqssr 3114	A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> B = A )   &    |- B C_ C   =>    |- ( ph -> A C_ C )
Theorem	eqimss 3115	Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A = B -> A C_ B )
Theorem	eqimss2 3116	Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
|- ( B = A -> A C_ B )
Theorem	eqimssi 3117	Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
|- A = B   =>    |- A C_ B
Theorem	eqimss2i 3118	Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
|- A = B   =>    |- B C_ A
Theorem	nssne1 3119	Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
|- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )
Theorem	nssne2 3120	Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
|- ( ( A C_ C /\ -. B C_ C ) -> A =/= B )
Theorem	nssr 3121*	Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
|- ( E. x ( x e. A /\ -. x e. B ) -> -. A C_ B )
Theorem	nelss 3122	Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
|- ( ( A e. B /\ -. A e. C ) -> -. B C_ C )
Theorem	ssrexf 3123	Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )
Theorem	ssrmof 3124	"At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
|- F/_ x A   &    |- F/_ x B   =>    |- ( A C_ B -> ( E* x e. B ph -> E* x e. A ph ) )
Theorem	ssralv 3125*	Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
|- ( A C_ B -> ( A. x e. B ph -> A. x e. A ph ) )
Theorem	ssrexv 3126*	Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
|- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )
Theorem	ralss 3127*	Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( A C_ B -> ( A. x e. A ph <-> A. x e. B ( x e. A -> ph ) ) )
Theorem	rexss 3128*	Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( A C_ B -> ( E. x e. A ph <-> E. x e. B ( x e. A /\ ph ) ) )
Theorem	ss2ab 3129	Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
|- ( { x | ph } C_ { x | ps } <-> A. x ( ph -> ps ) )
Theorem	abss 3130*	Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
|- ( { x | ph } C_ A <-> A. x ( ph -> x e. A ) )
Theorem	ssab 3131*	Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
|- ( A C_ { x | ph } <-> A. x ( x e. A -> ph ) )
Theorem	ssabral 3132*	The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
|- ( A C_ { x | ph } <-> A. x e. A ph )
Theorem	ss2abi 3133	Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
|- ( ph -> ps )   =>    |- { x | ph } C_ { x | ps }
Theorem	ss2abdv 3134*	Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> { x | ps } C_ { x | ch } )
Theorem	abssdv 3135*	Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
|- ( ph -> ( ps -> x e. A ) )   =>    |- ( ph -> { x | ps } C_ A )
Theorem	abssi 3136*	Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
|- ( ph -> x e. A )   =>    |- { x | ph } C_ A
Theorem	ss2rab 3137	Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
|- ( { x e. A | ph } C_ { x e. A | ps } <-> A. x e. A ( ph -> ps ) )
Theorem	rabss 3138*	Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
|- ( { x e. A | ph } C_ B <-> A. x e. A ( ph -> x e. B ) )
Theorem	ssrab 3139*	Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
|- ( B C_ { x e. A | ph } <-> ( B C_ A /\ A. x e. B ph ) )
Theorem	ssrabdv 3140*	Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
|- ( ph -> B C_ A )   &    |- ( ( ph /\ x e. B ) -> ps )   =>    |- ( ph -> B C_ { x e. A | ps } )
Theorem	rabssdv 3141*	Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)
|- ( ( ph /\ x e. A /\ ps ) -> x e. B )   =>    |- ( ph -> { x e. A | ps } C_ B )
Theorem	ss2rabdv 3142*	Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )   =>    |- ( ph -> { x e. A | ps } C_ { x e. A | ch } )
Theorem	ss2rabi 3143	Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
|- ( x e. A -> ( ph -> ps ) )   =>    |- { x e. A | ph } C_ { x e. A | ps }
Theorem	rabss2 3144*	Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A C_ B -> { x e. A | ph } C_ { x e. B | ph } )
Theorem	ssab2 3145*	Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
|- { x | ( x e. A /\ ph ) } C_ A
Theorem	ssrab2 3146*	Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
|- { x e. A | ph } C_ A
Theorem	ssrabeq 3147*	If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
|- ( V C_ { x e. V | ph } <-> V = { x e. V | ph } )
Theorem	rabssab 3148	A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- { x e. A | ph } C_ { x | ph }
Theorem	uniiunlem 3149*	A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
|- ( A. x e. A B e. D -> ( A. x e. A B e. C <-> { y | E. x e. A y = B } C_ C ) )
2.1.13  The difference, union, and intersection of two classes
2.1.13.1  The difference of two classes
Theorem	dfdif3 3150*	Alternate definition of class difference. Definition of relative set complement in Section 2.3 of [Pierik], p. 10. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
|- ( A \ B ) = { x e. A | A. y e. B x =/= y }
Theorem	difeq1 3151	Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A = B -> ( A \ C ) = ( B \ C ) )
Theorem	difeq2 3152	Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A = B -> ( C \ A ) = ( C \ B ) )
Theorem	difeq12 3153	Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
|- ( ( A = B /\ C = D ) -> ( A \ C ) = ( B \ D ) )
Theorem	difeq1i 3154	Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
|- A = B   =>    |- ( A \ C ) = ( B \ C )
Theorem	difeq2i 3155	Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
|- A = B   =>    |- ( C \ A ) = ( C \ B )
Theorem	difeq12i 3156	Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
|- A = B   &    |- C = D   =>    |- ( A \ C ) = ( B \ D )
Theorem	difeq1d 3157	Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A \ C ) = ( B \ C ) )
Theorem	difeq2d 3158	Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C \ A ) = ( C \ B ) )
Theorem	difeq12d 3159	Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A \ C ) = ( B \ D ) )
Theorem	difeqri 3160*	Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( x e. A /\ -. x e. B ) <-> x e. C )   =>    |- ( A \ B ) = C
Theorem	nfdif 3161	Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A \ B )
Theorem	eldifi 3162	Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
|- ( A e. ( B \ C ) -> A e. B )
Theorem	eldifn 3163	Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
|- ( A e. ( B \ C ) -> -. A e. C )
Theorem	elndif 3164	A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
|- ( A e. B -> -. A e. ( C \ B ) )
Theorem	difdif 3165	Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
|- ( A \ ( B \ A ) ) = A
Theorem	difss 3166	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
|- ( A \ B ) C_ A
Theorem	difssd 3167	A difference of two classes is contained in the minuend. Deduction form of difss 3166. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> ( A \ B ) C_ A )
Theorem	difss2 3168	If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
|- ( A C_ ( B \ C ) -> A C_ B )
Theorem	difss2d 3169	If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3168. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ ( B \ C ) )   =>    |- ( ph -> A C_ B )
Theorem	ssdifss 3170	Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
|- ( A C_ B -> ( A \ C ) C_ B )
Theorem	ddifnel 3171*	Double complement under universal class. The hypothesis corresponds to stability of membership in A, which is weaker than decidability (see dcstab 812). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3172) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that A is a subset of _V \ ( _V \ A ), see ddifss 3278. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( -. x e. ( _V \ A ) -> x e. A )   =>    |- ( _V \ ( _V \ A ) ) = A
Theorem	ddifstab 3172*	A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
|- ( ( _V \ ( _V \ A ) ) = A <-> A. xSTAB x e. A )
Theorem	ssconb 3173	Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
|- ( ( A C_ C /\ B C_ C ) -> ( A C_ ( C \ B ) <-> B C_ ( C \ A ) ) )
Theorem	sscon 3174	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
|- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )
Theorem	ssdif 3175	Difference law for subsets. (Contributed by NM, 28-May-1998.)
|- ( A C_ B -> ( A \ C ) C_ ( B \ C ) )
Theorem	ssdifd 3176	If A is contained in B, then ( A \ C ) is contained in ( B \ C ). Deduction form of ssdif 3175. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( A \ C ) C_ ( B \ C ) )
Theorem	sscond 3177	If A is contained in B, then ( C \ B ) is contained in ( C \ A ). Deduction form of sscon 3174. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C \ B ) C_ ( C \ A ) )
Theorem	ssdifssd 3178	If A is contained in B, then ( A \ C ) is also contained in B. Deduction form of ssdifss 3170. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( A \ C ) C_ B )
Theorem	ssdif2d 3179	If A is contained in B and C is contained in D, then ( A \ D ) is contained in ( B \ C ). Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A C_ B )   &    |- ( ph -> C C_ D )   =>    |- ( ph -> ( A \ D ) C_ ( B \ C ) )
Theorem	raldifb 3180	Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
|- ( A. x e. A ( x e/ B -> ph ) <-> A. x e. ( A \ B ) ph )
2.1.13.2  The union of two classes
Theorem	elun 3181	Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
|- ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) )
Theorem	uneqri 3182*	Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
|- ( ( x e. A \/ x e. B ) <-> x e. C )   =>    |- ( A u. B ) = C
Theorem	unidm 3183	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
|- ( A u. A ) = A
Theorem	uncom 3184	Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A u. B ) = ( B u. A )
Theorem	equncom 3185	If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)
|- ( A = ( B u. C ) <-> A = ( C u. B ) )
Theorem	equncomi 3186	Inference form of equncom 3185. (Contributed by Alan Sare, 18-Feb-2012.)
|- A = ( B u. C )   =>    |- A = ( C u. B )
Theorem	uneq1 3187	Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( A u. C ) = ( B u. C ) )
Theorem	uneq2 3188	Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C u. A ) = ( C u. B ) )
Theorem	uneq12 3189	Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
|- ( ( A = B /\ C = D ) -> ( A u. C ) = ( B u. D ) )
Theorem	uneq1i 3190	Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- ( A u. C ) = ( B u. C )
Theorem	uneq2i 3191	Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- ( C u. A ) = ( C u. B )
Theorem	uneq12i 3192	Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &    |- C = D   =>    |- ( A u. C ) = ( B u. D )
Theorem	uneq1d 3193	Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A u. C ) = ( B u. C ) )
Theorem	uneq2d 3194	Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C u. A ) = ( C u. B ) )
Theorem	uneq12d 3195	Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A u. C ) = ( B u. D ) )
Theorem	nfun 3196	Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A u. B )
Theorem	unass 3197	Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A u. B ) u. C ) = ( A u. ( B u. C ) )
Theorem	un12 3198	A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )
Theorem	un23 3199	A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A u. B ) u. C ) = ( ( A u. C ) u. B )
Theorem	un4 3200	A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
|- ( ( A u. B ) u. ( C u. D ) ) = ( ( A u. C ) u. ( B u. D ) )