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
Theorem	ineq1 3301	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 3302	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 3303	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 3304	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 3305	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 3306	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 3307	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 3308	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 3309	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 3310	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 3311	A frequently-used variant of subclass definition df-ss 3115. (Contributed by NM, 10-Jan-2015.)
|- ( A C_ B <-> ( B i^i A ) = A )
Theorem	dfss5 3312	Another definition of subclasshood. Similar to df-ss 3115, dfss 3116, and dfss1 3311. (Contributed by David Moews, 1-May-2017.)
|- ( A C_ B <-> A = ( B i^i A ) )
Theorem	nfin 3313	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 3314	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 3315*	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 3316	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 3317	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 3318	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 3319	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 3320	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 3321	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 3322	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 3323	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 3324	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 3325	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 3326	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 3327	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 3328	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 3329	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 3330	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 3331	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 3332	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 3333	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 3334	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 3335	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 3336	Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
|- ( A C_ C -> ( A i^i B ) C_ C )
Theorem	inss 3337	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 3338	Absorption law for union. (Contributed by NM, 16-Apr-2006.)
|- ( A u. ( A i^i B ) ) = A
Theorem	inabs 3339	Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
|- ( A i^i ( A u. B ) ) = A
Theorem	dfss4st 3340*	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 3341	Double complement and subset. Similar to ddifss 3345 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 3342	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 3343	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 3344	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 3345	Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3238), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
|- A C_ ( _V \ ( _V \ A ) )
Theorem	unssin 3346	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 3347	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 ) ) )
Theorem	inssddif 3348	Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
|- ( A i^i B ) C_ ( A \ ( A \ B ) )
Theorem	invdif 3349	Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
|- ( A i^i ( _V \ B ) ) = ( A \ B )
Theorem	indif 3350	Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
|- ( A i^i ( A \ B ) ) = ( A \ B )
Theorem	indif2 3351	Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
|- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )
Theorem	indif1 3352	Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
|- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )
Theorem	indifcom 3353	Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
|- ( A i^i ( B \ C ) ) = ( B i^i ( A \ C ) )
Theorem	indi 3354	Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
Theorem	undi 3355	Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( A u. ( B i^i C ) ) = ( ( A u. B ) i^i ( A u. C ) )
Theorem	indir 3356	Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
|- ( ( A u. B ) i^i C ) = ( ( A i^i C ) u. ( B i^i C ) )
Theorem	undir 3357	Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
|- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )
Theorem	uneqin 3358	Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( A u. B ) = ( A i^i B ) <-> A = B )
Theorem	difundi 3359	Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
|- ( A \ ( B u. C ) ) = ( ( A \ B ) i^i ( A \ C ) )
Theorem	difundir 3360	Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
|- ( ( A u. B ) \ C ) = ( ( A \ C ) u. ( B \ C ) )
Theorem	difindiss 3361	Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
|- ( ( A \ B ) u. ( A \ C ) ) C_ ( A \ ( B i^i C ) )
Theorem	difindir 3362	Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
|- ( ( A i^i B ) \ C ) = ( ( A \ C ) i^i ( B \ C ) )
Theorem	indifdir 3363	Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
|- ( ( A \ B ) i^i C ) = ( ( A i^i C ) \ ( B i^i C ) )
Theorem	difdif2ss 3364	Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
|- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) )
Theorem	undm 3365	De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
|- ( _V \ ( A u. B ) ) = ( ( _V \ A ) i^i ( _V \ B ) )
Theorem	indmss 3366	De Morgan's law for intersection. In classical logic, this would be equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19. (Contributed by Jim Kingdon, 27-Jul-2018.)
|- ( ( _V \ A ) u. ( _V \ B ) ) C_ ( _V \ ( A i^i B ) )
Theorem	difun1 3367	A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
|- ( A \ ( B u. C ) ) = ( ( A \ B ) \ C )
Theorem	undif3ss 3368	A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- ( A u. ( B \ C ) ) C_ ( ( A u. B ) \ ( C \ A ) )
Theorem	difin2 3369	Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )
Theorem	dif32 3370	Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
|- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )
Theorem	difabs 3371	Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
|- ( ( A \ B ) \ B ) = ( A \ B )
Theorem	symdif1 3372	Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
|- ( ( A \ B ) u. ( B \ A ) ) = ( ( A u. B ) \ ( A i^i B ) )
2.1.13.5  Class abstractions with difference, union, and intersection of two classes
Theorem	symdifxor 3373*	Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
|- ( ( A \ B ) u. ( B \ A ) ) = { x | ( x e. A \/_ x e. B ) }
Theorem	unab 3374	Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( { x | ph } u. { x | ps } ) = { x | ( ph \/ ps ) }
Theorem	inab 3375	Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( { x | ph } i^i { x | ps } ) = { x | ( ph /\ ps ) }
Theorem	difab 3376	Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) }
Theorem	notab 3377	A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
|- { x | -. ph } = ( _V \ { x | ph } )
Theorem	unrab 3378	Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
|- ( { x e. A | ph } u. { x e. A | ps } ) = { x e. A | ( ph \/ ps ) }
Theorem	inrab 3379	Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
|- ( { x e. A | ph } i^i { x e. A | ps } ) = { x e. A | ( ph /\ ps ) }
Theorem	inrab2 3380*	Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
|- ( { x e. A | ph } i^i B ) = { x e. ( A i^i B ) | ph }
Theorem	difrab 3381	Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
|- ( { x e. A | ph } \ { x e. A | ps } ) = { x e. A | ( ph /\ -. ps ) }
Theorem	dfrab2 3382*	Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
|- { x e. A | ph } = ( { x | ph } i^i A )
Theorem	dfrab3 3383*	Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
|- { x e. A | ph } = ( A i^i { x | ph } )
Theorem	notrab 3384*	Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
|- ( A \ { x e. A | ph } ) = { x e. A | -. ph }
Theorem	dfrab3ss 3385*	Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
|- ( A C_ B -> { x e. A | ph } = ( A i^i { x e. B | ph } ) )
Theorem	rabun2 3386	Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
|- { x e. ( A u. B ) | ph } = ( { x e. A | ph } u. { x e. B | ph } )
2.1.13.6  Restricted uniqueness with difference, union, and intersection
Theorem	reuss2 3387*	Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
|- ( ( ( A C_ B /\ A. x e. A ( ph -> ps ) ) /\ ( E. x e. A ph /\ E! x e. B ps ) ) -> E! x e. A ph )
Theorem	reuss 3388*	Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
|- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )
Theorem	reuun1 3389*	Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
|- ( ( E. x e. A ph /\ E! x e. ( A u. B ) ( ph \/ ps ) ) -> E! x e. A ph )
Theorem	reuun2 3390*	Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
|- ( -. E. x e. B ph -> ( E! x e. ( A u. B ) ph <-> E! x e. A ph ) )
Theorem	reupick 3391*	Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
|- ( ( ( A C_ B /\ ( E. x e. A ph /\ E! x e. B ph ) ) /\ ph ) -> ( x e. A <-> x e. B ) )
Theorem	reupick3 3392*	Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
|- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )
Theorem	reupick2 3393*	Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
|- ( ( ( A. x e. A ( ps -> ph ) /\ E. x e. A ps /\ E! x e. A ph ) /\ x e. A ) -> ( ph <-> ps ) )
2.1.14  The empty set
Syntax	c0 3394	Extend class notation to include the empty set.
class (/)
Definition	df-nul 3395	Define the empty set. Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 3396. (Contributed by NM, 5-Aug-1993.)
|- (/) = ( _V \ _V )
Theorem	dfnul2 3396	Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
|- (/) = { x | -. x = x }
Theorem	dfnul3 3397	Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
|- (/) = { x e. A | -. x e. A }
Theorem	noel 3398	The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
|- -. A e. (/)
Theorem	n0i 3399	If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2728. (Contributed by NM, 31-Dec-1993.)
|- ( B e. A -> -. A = (/) )
Theorem	ne0i 3400	If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2728. (Contributed by NM, 31-Dec-1993.)
|- ( B e. A -> A =/= (/) )