P. 34 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)

Type	Label	Description
Statement
2.1.13.2  The union of two classes
Theorem	elun 3301	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 3302*	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 3303	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
|- ( A u. A ) = A
Theorem	uncom 3304	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 3305	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 3306	Inference form of equncom 3305. (Contributed by Alan Sare, 18-Feb-2012.)
|- A = ( B u. C )   =>    |- A = ( C u. B )
Theorem	uneq1 3307	Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( A u. C ) = ( B u. C ) )
Theorem	uneq2 3308	Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C u. A ) = ( C u. B ) )
Theorem	uneq12 3309	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 3310	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 3311	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 3312	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 3313	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 3314	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 3315	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 3316	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 3317	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 3318	A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|- ( A u. ( B u. C ) ) = ( B u. ( A u. C ) )
Theorem	un23 3319	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 3320	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 ) )
Theorem	unundi 3321	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|- ( A u. ( B u. C ) ) = ( ( A u. B ) u. ( A u. C ) )
Theorem	unundir 3322	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|- ( ( A u. B ) u. C ) = ( ( A u. C ) u. ( B u. C ) )
Theorem	ssun1 3323	Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
|- A C_ ( A u. B )
Theorem	ssun2 3324	Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
|- A C_ ( B u. A )
Theorem	ssun3 3325	Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
|- ( A C_ B -> A C_ ( B u. C ) )
Theorem	ssun4 3326	Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
|- ( A C_ B -> A C_ ( C u. B ) )
Theorem	elun1 3327	Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
|- ( A e. B -> A e. ( B u. C ) )
Theorem	elun2 3328	Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
|- ( A e. B -> A e. ( C u. B ) )
Theorem	unss1 3329	Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
Theorem	ssequn1 3330	A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A C_ B <-> ( A u. B ) = B )
Theorem	unss2 3331	Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B -> ( C u. A ) C_ ( C u. B ) )
Theorem	unss12 3332	Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
|- ( ( A C_ B /\ C C_ D ) -> ( A u. C ) C_ ( B u. D ) )
Theorem	ssequn2 3333	A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
|- ( A C_ B <-> ( B u. A ) = B )
Theorem	unss 3334	The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
|- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
Theorem	unssi 3335	An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
|- A C_ C   &    |- B C_ C   =>    |- ( A u. B ) C_ C
Theorem	unssd 3336	A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A C_ C )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> ( A u. B ) C_ C )
Theorem	unssad 3337	If ( A u. B ) is contained in C, so is A. One-way deduction form of unss 3334. Partial converse of unssd 3336. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> ( A u. B ) C_ C )   =>    |- ( ph -> A C_ C )
Theorem	unssbd 3338	If ( A u. B ) is contained in C, so is B. One-way deduction form of unss 3334. Partial converse of unssd 3336. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> ( A u. B ) C_ C )   =>    |- ( ph -> B C_ C )
Theorem	ssun 3339	A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
|- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) )
Theorem	rexun 3340	Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
|- ( E. x e. ( A u. B ) ph <-> ( E. x e. A ph \/ E. x e. B ph ) )
Theorem	ralunb 3341	Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A. x e. ( A u. B ) ph <-> ( A. x e. A ph /\ A. x e. B ph ) )
Theorem	ralun 3342	Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( A. x e. A ph /\ A. x e. B ph ) -> A. x e. ( A u. B ) ph )
2.1.13.3  The intersection of two classes
Theorem	elin 3343	Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
|- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
Theorem	elini 3344	Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- A e. B   &    |- A e. C   =>    |- A e. ( B i^i C )
Theorem	elind 3345	Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> X e. A )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X e. ( A i^i B ) )
Theorem	elinel1 3346	Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. ( B i^i C ) -> A e. B )
Theorem	elinel2 3347	Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. ( B i^i C ) -> A e. C )
Theorem	elin2 3348	Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
|- X = ( B i^i C )   =>    |- ( A e. X <-> ( A e. B /\ A e. C ) )
Theorem	elin1d 3349	Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
|- ( ph -> X e. ( A i^i B ) )   =>    |- ( ph -> X e. A )
Theorem	elin2d 3350	Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
|- ( ph -> X e. ( A i^i B ) )   =>    |- ( ph -> X e. B )
Theorem	elin3 3351	Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
|- X = ( ( B i^i C ) i^i D )   =>    |- ( A e. X <-> ( A e. B /\ A e. C /\ A e. D ) )
Theorem	incom 3352	Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
|- ( A i^i B ) = ( B i^i A )
Theorem	ineqri 3353*	Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
|- ( ( x e. A /\ x e. B ) <-> x e. C )   =>    |- ( A i^i B ) = C
Theorem	ineq1 3354	Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
|- ( A = B -> ( A i^i C ) = ( B i^i C ) )
Theorem	ineq2 3355	Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
|- ( A = B -> ( C i^i A ) = ( C i^i B ) )
Theorem	ineq12 3356	Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
|- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )
Theorem	ineq1i 3357	Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
|- A = B   =>    |- ( A i^i C ) = ( B i^i C )
Theorem	ineq2i 3358	Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
|- A = B   =>    |- ( C i^i A ) = ( C i^i B )
Theorem	ineq12i 3359	Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &    |- C = D   =>    |- ( A i^i C ) = ( B i^i D )
Theorem	ineq1d 3360	Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A i^i C ) = ( B i^i C ) )
Theorem	ineq2d 3361	Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C i^i A ) = ( C i^i B ) )
Theorem	ineq12d 3362	Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A i^i C ) = ( B i^i D ) )
Theorem	ineqan12d 3363	Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A i^i C ) = ( B i^i D ) )
Theorem	dfss1 3364	A frequently-used variant of subclass definition df-ss 3167. (Contributed by NM, 10-Jan-2015.)
|- ( A C_ B <-> ( B i^i A ) = A )
Theorem	dfss5 3365	Another definition of subclasshood. Similar to df-ss 3167, dfss 3168, and dfss1 3364. (Contributed by David Moews, 1-May-2017.)
|- ( A C_ B <-> A = ( B i^i A ) )
Theorem	nfin 3366	Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A i^i B )
Theorem	csbing 3367	Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
|- ( A e. B -> [_ A / x ]_ ( C i^i D ) = ( [_ A / x ]_ C i^i [_ A / x ]_ D ) )
Theorem	rabbi2dva 3368*	Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
|- ( ( ph /\ x e. A ) -> ( x e. B <-> ps ) )   =>    |- ( ph -> ( A i^i B ) = { x e. A | ps } )
Theorem	inidm 3369	Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
|- ( A i^i A ) = A
Theorem	inass 3370	Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
|- ( ( A i^i B ) i^i C ) = ( A i^i ( B i^i C ) )
Theorem	in12 3371	A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|- ( A i^i ( B i^i C ) ) = ( B i^i ( A i^i C ) )
Theorem	in32 3372	A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i B )
Theorem	in13 3373	A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|- ( A i^i ( B i^i C ) ) = ( C i^i ( B i^i A ) )
Theorem	in31 3374	A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|- ( ( A i^i B ) i^i C ) = ( ( C i^i B ) i^i A )
Theorem	inrot 3375	Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
|- ( ( A i^i B ) i^i C ) = ( ( C i^i A ) i^i B )
Theorem	in4 3376	Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
|- ( ( A i^i B ) i^i ( C i^i D ) ) = ( ( A i^i C ) i^i ( B i^i D ) )
Theorem	inindi 3377	Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
|- ( A i^i ( B i^i C ) ) = ( ( A i^i B ) i^i ( A i^i C ) )
Theorem	inindir 3378	Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
|- ( ( A i^i B ) i^i C ) = ( ( A i^i C ) i^i ( B i^i C ) )
Theorem	sseqin2 3379	A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
|- ( A C_ B <-> ( B i^i A ) = A )
Theorem	inss1 3380	The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
|- ( A i^i B ) C_ A
Theorem	inss2 3381	The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
|- ( A i^i B ) C_ B
Theorem	ssin 3382	Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A C_ B /\ A C_ C ) <-> A C_ ( B i^i C ) )
Theorem	ssini 3383	An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
|- A C_ B   &    |- A C_ C   =>    |- A C_ ( B i^i C )
Theorem	ssind 3384	A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A C_ B )   &    |- ( ph -> A C_ C )   =>    |- ( ph -> A C_ ( B i^i C ) )
Theorem	ssrin 3385	Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
Theorem	sslin 3386	Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
|- ( A C_ B -> ( C i^i A ) C_ ( C i^i B ) )
Theorem	ssrind 3387	Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( A i^i C ) C_ ( B i^i C ) )
Theorem	ss2in 3388	Intersection of subclasses. (Contributed by NM, 5-May-2000.)
|- ( ( A C_ B /\ C C_ D ) -> ( A i^i C ) C_ ( B i^i D ) )
Theorem	ssinss1 3389	Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
|- ( A C_ C -> ( A i^i B ) C_ C )
Theorem	inss 3390	Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
|- ( ( A C_ C \/ B C_ C ) -> ( A i^i B ) C_ C )
2.1.13.4  Combinations of difference, union, and intersection of two classes
Theorem	unabs 3391	Absorption law for union. (Contributed by NM, 16-Apr-2006.)
|- ( A u. ( A i^i B ) ) = A
Theorem	inabs 3392	Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
|- ( A i^i ( A u. B ) ) = A
Theorem	dfss4st 3393*	Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A. xSTAB x e. A -> ( A C_ B <-> ( B \ ( B \ A ) ) = A ) )
Theorem	ssddif 3394	Double complement and subset. Similar to ddifss 3398 but inside a class B instead of the universal class _V. In classical logic the subset operation on the right hand side could be an equality (that is, A C_ B <-> ( B \ ( B \ A ) ) = A). (Contributed by Jim Kingdon, 24-Jul-2018.)
|- ( A C_ B <-> A C_ ( B \ ( B \ A ) ) )
Theorem	unssdif 3395	Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
|- ( A u. B ) C_ ( _V \ ( ( _V \ A ) \ B ) )
Theorem	inssdif 3396	Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
|- ( A i^i B ) C_ ( A \ ( _V \ B ) )
Theorem	difin 3397	Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A \ ( A i^i B ) ) = ( A \ B )
Theorem	ddifss 3398	Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3291), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
|- A C_ ( _V \ ( _V \ A ) )
Theorem	unssin 3399	Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
|- ( A u. B ) C_ ( _V \ ( ( _V \ A ) i^i ( _V \ B ) ) )
Theorem	inssun 3400	Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
|- ( A i^i B ) C_ ( _V \ ( ( _V \ A ) u. ( _V \ B ) ) )