P. 33 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cbvralcsf 3201	A more general version of cbvralf 2769 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 3202	A more general version of cbvrexf 2770 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 3203	A more general version of cbvreuv 2780 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 3204	A more general version of cbvrab 2811 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 3205*	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 3206*	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 3207*	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 3208	Extend class notation to include class difference (read: " A minus B").
class ( A \ B )
Syntax	cun 3209	Extend class notation to include union of two classes (read: " A union B").
class ( A u. B )
Syntax	cin 3210	Extend class notation to include the intersection of two classes (read: " A intersect B").
class ( A i^i B )
Syntax	wss 3211	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 3212*	Soundness justification theorem for df-dif 3213. (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 3213*	Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union ( A u. B ) (df-un 3215) and intersection ( A i^i B ) (df-in 3217). 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 3214*	Soundness justification theorem for df-un 3215. (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 3215*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference ( A \ B ) (df-dif 3213) and intersection ( A i^i B ) (df-in 3217). (Contributed by NM, 23-Aug-1993.)
|- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
Theorem	injust 3216*	Soundness justification theorem for df-in 3217. (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 3217*	Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union ( A u. B ) (df-un 3215) and difference ( A \ B ) (df-dif 3213). (Contributed by NM, 29-Apr-1994.)
|- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
Theorem	dfin5 3218*	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 3219*	Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)
|- ( A \ B ) = { x e. A | -. x e. B }
Theorem	eldif 3220	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 3221	If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3220. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. B )   &    |- ( ph -> -. A e. C )   =>    |- ( ph -> A e. ( B \ C ) )
Theorem	eldifad 3222	If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3220. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ( B \ C ) )   =>    |- ( ph -> A e. B )
Theorem	eldifbd 3223	If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3220. (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 3224	Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that A C_ A (proved in ssid 3258). For a more traditional definition, but requiring a dummy variable, see ssalel 3226. Other possible definitions are given by dfss3 3227, ssequn1 3389, ssequn2 3392, and sseqin2 3440. (Contributed by NM, 27-Apr-1994.)
|- ( A C_ B <-> ( A i^i B ) = A )
Theorem	dfss 3225	Variant of subclass definition df-ss 3224. (Contributed by NM, 3-Sep-2004.)
|- ( A C_ B <-> A = ( A i^i B ) )
Theorem	ssalel 3226*	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 3227*	Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
|- ( A C_ B <-> A. x e. A x e. B )
Theorem	dfss2 3228	Alternate definition of the subclass relationship between two classes. Exercise 9 of [TakeutiZaring] p. 18. This is another name for df-ss 3224 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 3229	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 3230	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 3231	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 3232	Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- ( A C_ B -> ( C e. A -> C e. B ) )
Theorem	ssel2 3233	Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
|- ( ( A C_ B /\ C e. A ) -> C e. B )
Theorem	sseli 3234	Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
|- A C_ B   =>    |- ( C e. A -> C e. B )
Theorem	sselii 3235	Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
|- A C_ B   &    |- C e. A   =>    |- C e. B
Theorem	sselid 3236	Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
|- A C_ B   &    |- ( ph -> C e. A )   =>    |- ( ph -> C e. B )
Theorem	sseld 3237	Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ( C e. A -> C e. B ) )
Theorem	sselda 3238	Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
|- ( ph -> A C_ B )   =>    |- ( ( ph /\ C e. A ) -> C e. B )
Theorem	sseldd 3239	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 3240	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 3241	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 )
Theorem	ssriv 3242*	Inference based on subclass definition. (Contributed by NM, 5-Aug-1993.)
|- ( x e. A -> x e. B )   =>    |- A C_ B
Theorem	ssrd 3243	Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   &    |- ( ph -> ( x e. A -> x e. B ) )   =>    |- ( ph -> A C_ B )
Theorem	ssrdv 3244*	Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)
|- ( ph -> ( x e. A -> x e. B ) )   =>    |- ( ph -> A C_ B )
Theorem	sstr2 3245	Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( A C_ B -> ( B C_ C -> A C_ C ) )
Theorem	sstr 3246	Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
|- ( ( A C_ B /\ B C_ C ) -> A C_ C )
Theorem	sstri 3247	Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
|- A C_ B   &    |- B C_ C   =>    |- A C_ C
Theorem	sstrd 3248	Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sstrid 3249	Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
|- A C_ B   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sstrdi 3250	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph -> A C_ B )   &    |- B C_ C   =>    |- ( ph -> A C_ C )
Theorem	sylan9ss 3251	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( ph -> A C_ B )   &    |- ( ps -> B C_ C )   =>    |- ( ( ph /\ ps ) -> A C_ C )
Theorem	sylan9ssr 3252	A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
|- ( ph -> A C_ B )   &    |- ( ps -> B C_ C )   =>    |- ( ( ps /\ ph ) -> A C_ C )
Theorem	eqss 3253	The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
|- ( A = B <-> ( A C_ B /\ B C_ A ) )
Theorem	eqssi 3254	Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
|- A C_ B   &    |- B C_ A   =>    |- A = B
Theorem	eqssd 3255	Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> B C_ A )   =>    |- ( ph -> A = B )
Theorem	eqrd 3256	Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   &    |- ( ph -> ( x e. A <-> x e. B ) )   =>    |- ( ph -> A = B )
Theorem	eqelssd 3257*	Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ x e. B ) -> x e. A )   =>    |- ( ph -> A = B )
Theorem	ssid 3258	Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- A C_ A
Theorem	ssidd 3259	Weakening of ssid 3258. (Contributed by BJ, 1-Sep-2022.)
|- ( ph -> A C_ A )
Theorem	ssv 3260	Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
|- A C_ _V
Theorem	sseq1 3261	Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
|- ( A = B -> ( A C_ C <-> B C_ C ) )
Theorem	sseq2 3262	Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
|- ( A = B -> ( C C_ A <-> C C_ B ) )
Theorem	sseq12 3263	Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
|- ( ( A = B /\ C = D ) -> ( A C_ C <-> B C_ D ) )
Theorem	sseq1i 3264	An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
|- A = B   =>    |- ( A C_ C <-> B C_ C )
Theorem	sseq2i 3265	An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
|- A = B   =>    |- ( C C_ A <-> C C_ B )
Theorem	sseq12i 3266	An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|- A = B   &    |- C = D   =>    |- ( A C_ C <-> B C_ D )
Theorem	sseq1d 3267	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A C_ C <-> B C_ C ) )
Theorem	sseq2d 3268	An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C C_ A <-> C C_ B ) )
Theorem	sseq12d 3269	An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A C_ C <-> B C_ D ) )
Theorem	eqsstri 3270	Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
|- A = B   &    |- B C_ C   =>    |- A C_ C
Theorem	eqsstrri 3271	Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
|- B = A   &    |- B C_ C   =>    |- A C_ C
Theorem	sseqtri 3272	Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
|- A C_ B   &    |- B = C   =>    |- A C_ C
Theorem	sseqtrri 3273	Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
|- A C_ B   &    |- C = B   =>    |- A C_ C
Theorem	eqsstrd 3274	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A = B )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	eqsstrrd 3275	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> B = A )   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sseqtrd 3276	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> B = C )   =>    |- ( ph -> A C_ C )
Theorem	sseqtrrd 3277	Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- ( ph -> C = B )   =>    |- ( ph -> A C_ C )
Theorem	3sstr3i 3278	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 3279	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 3280	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 3281	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 3282	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 3283	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	eqsstrid 3284	B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- A = B   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	eqsstrrid 3285	B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- B = A   &    |- ( ph -> B C_ C )   =>    |- ( ph -> A C_ C )
Theorem	sseqtrdi 3286	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- B = C   =>    |- ( ph -> A C_ C )
Theorem	sseqtrrdi 3287	A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
|- ( ph -> A C_ B )   &    |- C = B   =>    |- ( ph -> A C_ C )
Theorem	sseqtrid 3288	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- B C_ A   &    |- ( ph -> A = C )   =>    |- ( ph -> B C_ C )
Theorem	sseqtrrid 3289	Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- B C_ A   &    |- ( ph -> C = A )   =>    |- ( ph -> B C_ C )
Theorem	eqsstrdi 3290	A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> A = B )   &    |- B C_ C   =>    |- ( ph -> A C_ C )
Theorem	eqsstrrdi 3291	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 3292	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 3293	Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
|- ( B = A -> A C_ B )
Theorem	eqimssi 3294	Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
|- A = B   =>    |- A C_ B
Theorem	eqimss2i 3295	Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
|- A = B   =>    |- B C_ A
Theorem	nssne1 3296	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 3297	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 3298*	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 3299	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 3300	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 ) )