P. 44 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vn0ALT 4301	Alternate proof of vn0 4299. Shorter, but requiring df-clel 2838, ax-8 2145. (Contributed by NM, 11-Sep-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ V ≠ ∅
Theorem	eq0f 4302	A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	neq0f 4303	A class is not empty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 4307 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	n0f 4304	A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 4308 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	eq0 4305*	A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2192, ax-12 2213. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2145, df-clel 2838. (Revised by GG, 6-Sep-2024.)
⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	eq0ALT 4306*	Alternate proof of eq0 4305. Shorter, but requiring df-clel 2838, ax-8 2145. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2192, ax-12 2213. (Revised by GG and Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
Theorem	neq0 4307*	A class is not empty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 21-Jun-1993.) Avoid ax-11 2192, ax-12 2213. (Revised by GG, 28-Jun-2024.)
⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	n0 4308*	A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 29-Sep-2006.) Avoid ax-11 2192, ax-12 2213. (Revised by GG, 28-Jun-2024.)
⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
Theorem	n0limd 4309*	Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	nel0 4310*	From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
⊢ ¬ 𝑥 ∈ 𝐴    ⇒   ⊢ 𝐴 = ∅
Theorem	reximdva0 4311*	Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	rspn0 4312*	Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2176, ax-12 2213. (Revised by GG, 28-Jun-2024.)
⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝜑 → 𝜑))
Theorem	n0rex 4313*	There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐴)
Theorem	ssn0rex 4314*	There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐵 𝑥 ∈ 𝐴)
Theorem	n0moeu 4315*	A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))
Theorem	rex0 4316	Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
⊢ ¬ ∃𝑥 ∈ ∅ 𝜑
Theorem	reu0 4317	Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
⊢ ¬ ∃!𝑥 ∈ ∅ 𝜑
Theorem	rmo0 4318	Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
⊢ ∃*𝑥 ∈ ∅ 𝜑
Theorem	0el 4319*	Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	n0el 4320*	Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
⊢ (¬ ∅ ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑢 𝑢 ∈ 𝑥)
Theorem	eqeuel 4321*	A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴)
Theorem	ssdif0 4322	Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∖ 𝐵) = ∅)
Theorem	difn0 4323	If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
⊢ ((𝐴 ∖ 𝐵) ≠ ∅ → 𝐴 ≠ 𝐵)
Theorem	pssdifn0 4324	A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) → (𝐵 ∖ 𝐴) ≠ ∅)
Theorem	pssdif 4325	A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
Theorem	ndisj 4326*	Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	inn0f 4327	A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	inn0 4328*	A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
Theorem	difin0ss 4329	Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
⊢ (((𝐴 ∖ 𝐵) ∩ 𝐶) = ∅ → (𝐶 ⊆ 𝐴 → 𝐶 ⊆ 𝐵))
Theorem	inssdif0 4330	Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) (Proof shortened by BJ, 18-Jul-2026.)
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
Theorem	inssdif0OLD 4331	Obsolete version of inssdif0 4330 as of 18-Jul-2026. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
Theorem	inindif 4332	The intersection and class difference of a class with another class are disjoint. With inundif 4441, this shows that such intersection and class difference partition the class 𝐴. (Contributed by Thierry Arnoux, 13-Sep-2017.)
⊢ ((𝐴 ∩ 𝐶) ∩ (𝐴 ∖ 𝐶)) = ∅
Theorem	difid 4333	The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.) (Revised by David Abernethy, 17-Jun-2012.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	difidALT 4334	Alternate proof of difid 4333. Shorter, but requiring ax-8 2145, df-clel 2838. (Contributed by NM, 22-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∖ 𝐴) = ∅
Theorem	dif0 4335	The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∖ ∅) = 𝐴
Theorem	ab0w 4336*	The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ab0 4337 using implicit substitution, which requires fewer axioms. (Contributed by GG, 3-Oct-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓)
Theorem	ab0 4337	The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4342 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2959). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2838, ax-8 2145. (Revised by GG, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.)
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Theorem	ab0ALT 4338	Alternate proof of ab0 4337, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Theorem	dfnf5 4339	Characterization of nonfreeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅))
Theorem	ab0orv 4340*	The class abstraction defined by a formula not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) Reduce axiom usage. (Revised by GG, 30-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)
Theorem	ab0orvALT 4341*	Alternate proof of ab0orv 4340, shorter but using more axioms. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)
Theorem	abn0 4342	Nonempty class abstraction. See also ab0 4337. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2838, ax-8 2145. (Revised by GG, 30-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
Theorem	rab0 4343	Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Theorem	rab0OLD 4344	Obsolete version of rab0 4343 as of 10-Jun-2026. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅
Theorem	rabeq0w 4345*	Condition for a restricted class abstraction to be empty. Version of rabeq0 4346 using implicit substitution, which does not require ax-10 2176, ax-11 2192, ax-12 2213, but requires ax-8 2145. (Contributed by GG, 30-Sep-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
Theorem	rabeq0 4346	Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	rabn0 4347	Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rabxm 4348*	Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
Theorem	rabnc 4349	Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ∅
Theorem	elneldisj 4350*	The set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element are disjoint. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}    &   ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶}    ⇒   ⊢ (𝐸 ∩ 𝑁) = ∅
Theorem	elnelun 4351*	The union of the set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}    &   ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶}    ⇒   ⊢ (𝐸 ∪ 𝑁) = 𝐴
Theorem	un0 4352	The union of a class with the empty set is itself. Dual of inv1 4356. Commuted form of 0un 4354. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 ∪ ∅) = 𝐴
Theorem	in0 4353	The intersection of a class with the empty set is the empty set. Dual of unv 4357. Commuted form of in0 4353. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ∩ ∅) = ∅
Theorem	0un 4354	The union of the empty set with a class is itself. Commuted form of un0 4352. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (∅ ∪ 𝐴) = 𝐴
Theorem	0in 4355	The intersection of the empty set with a class is the empty set. Commuted form of 0in 4355. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (∅ ∩ 𝐴) = ∅
Theorem	inv1 4356	The intersection of a class with the universal class is itself. Dual of un0 4352. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∩ V) = 𝐴
Theorem	unv 4357	The union of a class with the universal class is the universal class. Dual of in0 4353. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∪ V) = V
Theorem	0ss 4358	The empty set is a subset of any class. Dual of ssv 3962. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
⊢ ∅ ⊆ 𝐴
Theorem	ss0b 4359	Any subset of the empty set is empty. Dual of vss 4366. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
Theorem	ss0 4360	Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅)
Theorem	sseq0b 4361	The only subclass of the empty class is itself. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Strengthen sseq0 4362 to a biconditional. (Revised by BJ, 19-Jul-2026.)
⊢ (𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 = ∅))
Theorem	sseq0 4362	A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
Theorem	ssn0 4363	A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → 𝐵 ≠ ∅)
Theorem	0dif 4364	The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
⊢ (∅ ∖ 𝐴) = ∅
Theorem	un00 4365	Two classes are both empty if and only if their union is empty. Dual of vvin 4367. (Contributed by NM, 11-Aug-2004.)
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)
Theorem	vss 4366	Only the universal class has the universal class as a subclass. Dual of ss0b 4359. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (V ⊆ 𝐴 ↔ 𝐴 = V)
Theorem	vvin 4367	Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 4365. (Contributed by BJ, 12-Jul-2026.)
⊢ ((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)
Theorem	0pss 4368	The empty set is a proper subset of any nonempty set. Dual of pssv 4370. (Contributed by NM, 27-Feb-1996.)
⊢ (∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅)
Theorem	npss0 4369	No set is a proper subset of the empty set. Dual of nvpss 4371. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ ¬ 𝐴 ⊊ ∅
Theorem	pssv 4370	Any non-universal class is a proper subclass of the universal class. Dual of 0pss 4368. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V)
Theorem	nvpss 4371	No class strictly includes the universal class. Dual of npss0 4369. (Contributed by BJ, 12-Jul-2026.)
⊢ ¬ V ⊊ 𝐴
Theorem	abf 4372	A class abstraction determined by a false formula is empty. (Contributed by NM, 20-Jan-2012.) Avoid ax-8 2145, ax-10 2176, ax-11 2192, ax-12 2213. (Revised by GG, 30-Jun-2024.)
⊢ ¬ 𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = ∅
Theorem	eq0rdv 4373*	Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.) Avoid ax-8 2145, df-clel 2838. (Revised by GG, 6-Sep-2024.)
⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	eq0rdvALT 4374*	Alternate proof of eq0rdv 4373. Shorter, but requiring df-clel 2838, ax-8 2145. (Contributed by NM, 11-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	csbprc 4375	The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.) (Proof shortened by JJ, 27-Aug-2021.)
⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
Theorem	csb0 4376	The proper substitution of a class into the empty set is the empty set. (Contributed by NM, 18-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌∅ = ∅
Theorem	sbcel12 4377	Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
Theorem	sbceqg 4378	Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
Theorem	sbceqi 4379	Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
⊢ 𝐴 ∈ V    &   ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷    &   ⊢ ⦋𝐴 / 𝑥⦌𝐶 = 𝐸    ⇒   ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸)
Theorem	sbcnel12g 4380	Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶))
Theorem	sbcne12 4381	Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)
Theorem	sbcel1g 4382*	Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵 ∈ 𝐶, whereas the scope of ⦋𝐴 / 𝑥⦌ is the class 𝐵. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝐶))
Theorem	sbceq1g 4383*	Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
Theorem	sbcel2 4384*	Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)
Theorem	sbceq2g 4385*	Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
Theorem	csbcom 4386*	Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋𝐵 / 𝑦⦌⦋𝐴 / 𝑥⦌𝐶
Theorem	sbcnestgfw 4387*	Nest the composition of two substitutions. Version of sbcnestgf 4392 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Mario Carneiro, 11-Nov-2016.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
Theorem	csbnestgfw 4388*	Nest the composition of two substitutions. Version of csbnestgf 4393 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
Theorem	sbcnestgw 4389*	Nest the composition of two substitutions. Version of sbcnestg 4394 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
Theorem	csbnestgw 4390*	Nest the composition of two substitutions. Version of csbnestg 4395 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
Theorem	sbcco3gw 4391*	Composition of two substitutions. Version of sbcco3g 4396 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))
Theorem	sbcnestgf 4392	Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker sbcnestgfw 4387 when possible. (Contributed by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
Theorem	csbnestgf 4393	Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker csbnestgfw 4388 when possible. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
Theorem	sbcnestg 4394*	Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker sbcnestgw 4389 when possible. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
Theorem	csbnestg 4395*	Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker csbnestgw 4390 when possible. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
Theorem	sbcco3g 4396*	Composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker sbcco3gw 4391 when possible. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))
Theorem	csbco3g 4397*	Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷)
Theorem	csbnest1g 4398	Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶)
Theorem	csbidm 4399*	Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
Theorem	csbvarg 4400	The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴)