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Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremn0ii 4301 If a class has elements, then it is not empty. Inference associated with n0i 4298. (Contributed by BJ, 15-Jul-2021.)
𝐴𝐵        ¬ 𝐵 = ∅
 
Theoremne0ii 4302 If a class has elements, then it is nonempty. Inference associated with ne0i 4299. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴𝐵       𝐵 ≠ ∅
 
Theoremvn0 4303 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅
 
Theoremeq0f 4304 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.)
𝑥𝐴       (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremneq0f 4305 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 4308 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by BJ, 15-Jul-2021.)
𝑥𝐴       𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremn0f 4306 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 4309 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by NM, 17-Oct-2003.)
𝑥𝐴       (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremeq0 4307* 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.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremneq0 4308* 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.)
𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremn0 4309* 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.)
(𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremnel0 4310* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅
 
Theoremreximdva0 4311* Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
 
Theoremrspn0 4312* Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 
Theoremn0rex 4313* There is an element in a nonempty class which is an element of the class. (Contributed by AV, 17-Dec-2020.)
(𝐴 ≠ ∅ → ∃𝑥𝐴 𝑥𝐴)
 
Theoremssn0rex 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.)
((𝐴𝐵𝐴 ≠ ∅) → ∃𝑥𝐵 𝑥𝐴)
 
Theoremn0moeu 4315* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
 
Theoremrex0 4316 Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑
 
Theoremreu0 4317 Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
¬ ∃!𝑥 ∈ ∅ 𝜑
 
Theoremrmo0 4318 Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
∃*𝑥 ∈ ∅ 𝜑
 
Theorem0el 4319* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
 
Theoremn0el 4320* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
 
Theoremeqeuel 4321* A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥𝐴)
 
Theoremssdif0 4322 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = ∅)
 
Theoremdifn0 4323 If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
 
Theorempssdifn0 4324 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
 
Theorempssdif 4325 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 
Theoremndisj 4326* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdifin0ss 4327 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
(((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))
 
Theoreminssdif0 4328 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.)
((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)
 
Theoremdifid 4329 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.)
(𝐴𝐴) = ∅
 
TheoremdifidALT 4330 Alternate proof of difid 4329. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅
 
Theoremdif0 4331 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
 
Theoremab0 4332 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4335 (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 3017). (Contributed by BJ, 19-Mar-2021.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremdfnf5 4333 Characterization of non-freeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
 
Theoremab0orv 4334* The class builder of a wff 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.)
({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
 
Theoremabn0 4335 Nonempty class abstraction. See also ab0 4332. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
Theoremrab0 4336 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.)
{𝑥 ∈ ∅ ∣ 𝜑} = ∅
 
Theoremrabeq0 4337 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)
 
Theoremrabn0 4338 Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
 
Theoremrabxm 4339* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
 
Theoremrabnc 4340* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
 
Theoremelneldisj 4341* 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 = {𝑠𝐴𝐵𝐶}    &   𝑁 = {𝑠𝐴𝐵𝐶}       (𝐸𝑁) = ∅
 
Theoremelnelun 4342* 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 = {𝑠𝐴𝐵𝐶}    &   𝑁 = {𝑠𝐴𝐵𝐶}       (𝐸𝑁) = 𝐴
 
Theoremun0 4343 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 15-Jul-1993.)
(𝐴 ∪ ∅) = 𝐴
 
Theoremin0 4344 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 21-Jun-1993.)
(𝐴 ∩ ∅) = ∅
 
Theorem0un 4345 The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∪ 𝐴) = 𝐴
 
Theorem0in 4346 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∩ 𝐴) = ∅
 
Theoreminv1 4347 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(𝐴 ∩ V) = 𝐴
 
Theoremunv 4348 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(𝐴 ∪ V) = V
 
Theorem0ss 4349 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 21-Jun-1993.)
∅ ⊆ 𝐴
 
Theoremss0b 4350 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
(𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
 
Theoremss0 4351 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)
 
Theoremsseq0 4352 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
 
Theoremssn0 4353 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)
 
Theorem0dif 4354 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) = ∅
 
Theoremabf 4355 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ 𝜑       {𝑥𝜑} = ∅
 
Theoremeq0rdv 4356* Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)
 
Theoremcsbprc 4357 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 / 𝑥𝐵 = ∅)
 
Theoremcsb0 4358 The proper substitution of a class into the empty set is the empty set. (Contributed by NM, 18-Aug-2018.)
𝐴 / 𝑥∅ = ∅
 
Theoremsbcel12 4359 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremsbceqg 4360 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
 
Theoremsbceqi 4361 Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   𝐴 / 𝑥𝐵 = 𝐷    &   𝐴 / 𝑥𝐶 = 𝐸       ([𝐴 / 𝑥]𝐵 = 𝐶𝐷 = 𝐸)
 
Theoremsbcnel12g 4362 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremsbcne12 4363 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremsbcel1g 4364* Move proper substitution in and out of a membership relation. Note that the scope of [𝐴 / 𝑥] is the wff 𝐵𝐶, whereas the scope of 𝐴 / 𝑥 is the class 𝐵. (Contributed by NM, 10-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐶))
 
Theoremsbceq1g 4365* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐶))
 
Theoremsbcel2 4366* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶)
 
Theoremsbceq2g 4367* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐵 = 𝐴 / 𝑥𝐶))
 
Theoremcsbcom 4368* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐵 / 𝑦𝐴 / 𝑥𝐶
 
Theoremsbcnestgfw 4369* Version of sbcnestgf 4374 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestgfw 4370* Version of csbnestgf 4375 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcnestgw 4371* Version of sbcnestg 4376 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestgw 4372* Version of csbnestg 4377 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcco3gw 4373* Version of sbcco3g 4378 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 
Theoremsbcnestgf 4374 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestgf 4375 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcnestg 4376* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
Theoremcsbnestg 4377* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
Theoremsbcco3g 4378* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 
Theoremcsbco3g 4379* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
 
Theoremcsbnest1g 4380 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
 
Theoremcsbidm 4381* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 
Theoremcsbvarg 4382 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
(𝐴𝑉𝐴 / 𝑥𝑥 = 𝐴)
 
Theoremcsbvargi 4383 The proper substitution of a class for a setvar variable results in the class (if the class exists), in inference form of csbvarg 4382. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V       𝐴 / 𝑥𝑥 = 𝐴
 
Theoremsbccsb 4384* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝜑𝑦𝐴 / 𝑥{𝑦𝜑})
 
Theoremsbccsb2 4385 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝜑𝐴𝐴 / 𝑥{𝑥𝜑})
 
Theoremrspcsbela 4386* Special case related to rspsbc 3861. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝐶𝐷) → 𝐴 / 𝑥𝐶𝐷)
 
Theoremsbnfc2 4387* Two ways of expressing "𝑥 is (effectively) not free in 𝐴". (Contributed by Mario Carneiro, 14-Oct-2016.)
(𝑥𝐴 ↔ ∀𝑦𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
 
Theoremcsbab 4388* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 19-Aug-2018.)
𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑}
 
Theoremcsbun 4389 Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.) (Revised by NM, 13-Sep-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theoremcsbin 4390 Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)
𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 
Theorem2nreu 4391* If there are two different sets fulfilling a wff (by implicit substitution), then there is no unique set fulfilling the wff. (Contributed by AV, 20-Jun-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝑋𝐵𝑋𝐴𝐵) → ((𝜓𝜒) → ¬ ∃!𝑥𝑋 𝜑))
 
Theoremun00 4392 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
 
Theoremvss 4393 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(V ⊆ 𝐴𝐴 = V)
 
Theorem0pss 4394 The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
(∅ ⊊ 𝐴𝐴 ≠ ∅)
 
Theoremnpss0 4395 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
¬ 𝐴 ⊊ ∅
 
Theorempssv 4396 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
(𝐴 ⊊ V ↔ ¬ 𝐴 = V)
 
Theoremdisj 4397* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
 
Theoremdisjr 4398* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
 
Theoremdisj1 4399* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
 
Theoremreldisj 4400 Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))
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