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Theorem List for Metamath Proof Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdifab 4301 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∖ {𝑥𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}
 
Theoremabanssl 4302 A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑𝜓)} ⊆ {𝑓𝜑}
 
Theoremabanssr 4303 A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑𝜓)} ⊆ {𝑓𝜓}
 
Theoremnotabw 4304* A class abstraction defined by a negation. Version of notab 4305 using implicit substitution, which does not require ax-10 2138, ax-12 2172. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦𝜓})
 
Theoremnotab 4305 A class abstraction defined by a negation. (Contributed by FL, 18-Sep-2010.)
{𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
 
Theoremunrab 4306 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
 
Theoreminrab 4307 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
 
Theoreminrab2 4308* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 
Theoremdifrab 4309 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
 
Theoremdfrab3 4310* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
 
Theoremdfrab2 4311* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
{𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
 
Theoremnotrab 4312* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
 
Theoremdfrab3ss 4313* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
(𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))
 
Theoremrabun2 4314 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
{𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜑})
 
2.1.13.7  Restricted uniqueness with difference, union, and intersection
 
Theoremreuun2 4315 Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))
 
Theoremreuss2 4316* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)
 
Theoremreuss 4317* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
 
Theoremreuun1 4318* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
 
Theoremreupick 4319* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
 
Theoremreupick3 4320* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
 
Theoremreupick2 4321* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))
 
Theoremeuelss 4322* Transfer uniqueness of an element to a smaller subclass. (Contributed by AV, 14-Apr-2020.)
((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥 𝑥𝐴)
 
2.1.14  The empty set
 
Syntaxc0 4323 Extend class notation to include the empty set.
class
 
Definitiondf-nul 4324 Define the empty set. More precisely, we should write "empty class". It will be posited in ax-nul 5307 that an empty set exists. Then, by uniqueness among classes (eq0 4344, as opposed to the weaker uniqueness among sets, nulmo 2709), it will follow that is indeed a set (0ex 5308). Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 4326. (Contributed by NM, 17-Jun-1993.) Clarify that at this point, it is not established that it is a set. (Revised by BJ, 22-Sep-2022.)
∅ = (V ∖ V)
 
Theoremdfnul4 4325 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4326. (Revised by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ⊥}
 
Theoremdfnul2 4326 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2138, ax-11 2155, and ax-12 2172. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
Theoremdfnul3 4327 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
 
Theoremdfnul2OLD 4328 Obsolete version of dfnul2 4326 as of 23-Sep-2024. (Contributed by NM, 26-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
Theoremdfnul3OLD 4329 Obsolete version of dfnul4 4325 as of 23-Sep-2024. (Contributed by NM, 25-Mar-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
 
Theoremdfnul4OLD 4330 Obsolete version of dfnul4 4325 as of 23-Sep-2024. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = {𝑥 ∣ ⊥}
 
Theoremnoel 4331 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) Remove dependency on ax-10 2138, ax-11 2155, and ax-12 2172. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
¬ 𝐴 ∈ ∅
 
TheoremnoelOLD 4332 Obsolete version of noel 4331 as of 18-Sep-2024. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝐴 ∈ ∅
 
Theoremnel02 4333 The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
(𝐴 = ∅ → ¬ 𝐵𝐴)
 
Theoremn0i 4334 If a class has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)
 
Theoremne0i 4335 If a class has elements, then it is nonempty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)
 
Theoremne0d 4336 Deduction form of ne0i 4335. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐴)       (𝜑𝐴 ≠ ∅)
 
Theoremn0ii 4337 If a class has elements, then it is not empty. Inference associated with n0i 4334. (Contributed by BJ, 15-Jul-2021.)
𝐴𝐵        ¬ 𝐵 = ∅
 
Theoremne0ii 4338 If a class has elements, then it is nonempty. Inference associated with ne0i 4335. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴𝐵       𝐵 ≠ ∅
 
Theoremvn0 4339 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 6-Sep-2024.)
V ≠ ∅
 
Theoremvn0ALT 4340 Alternate proof of vn0 4339. Shorter, but requiring df-clel 2811, ax-8 2109. (Contributed by NM, 11-Sep-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
V ≠ ∅
 
Theoremeq0f 4341 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 4342 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 4346 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 4343 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 4347 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 4344* 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 2155, ax-12 2172. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 6-Sep-2024.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremeq0ALT 4345* Alternate proof of eq0 4344. Shorter, but requiring df-clel 2811, ax-8 2109. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2155, ax-12 2172. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremneq0 4346* 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 2155, ax-12 2172. (Revised by Gino Giotto, 28-Jun-2024.)
𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremn0 4347* 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 2155, ax-12 2172. (Revised by Gino Giotto, 28-Jun-2024.)
(𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremeq0OLDOLD 4348* Obsolete version of eq0 4344 as of 28-Jun-2024. (Contributed by NM, 29-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremneq0OLD 4349* Obsolete version of neq0 4346 as of 28-Jun-2024. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremn0OLD 4350* Obsolete version of n0 4347 as of 28-Jun-2024. (Contributed by NM, 29-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremnel0 4351* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅
 
Theoremreximdva0 4352* Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
 
Theoremrspn0 4353* Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2138, ax-12 2172. (Revised by Gino Giotto, 28-Jun-2024.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 
Theoremrspn0OLD 4354* Obsolete version of rspn0 4353 as of 28-Jun-2024. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 
Theoremn0rex 4355* 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 4356* 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 4357* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
 
Theoremrex0 4358 Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑
 
Theoremreu0 4359 Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
¬ ∃!𝑥 ∈ ∅ 𝜑
 
Theoremrmo0 4360 Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
∃*𝑥 ∈ ∅ 𝜑
 
Theorem0el 4361* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
 
Theoremn0el 4362* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
 
Theoremeqeuel 4363* 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 4364 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = ∅)
 
Theoremdifn0 4365 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 4366 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
 
Theorempssdif 4367 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 
Theoremndisj 4368* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
 
Theoremdifin0ss 4369 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
(((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))
 
Theoreminssdif0 4370 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 4371 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.)
(𝐴𝐴) = ∅
 
TheoremdifidALT 4372 Alternate proof of difid 4371. Shorter, but requiring ax-8 2109, df-clel 2811. (Contributed by NM, 22-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅
 
Theoremdif0 4373 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
 
Theoremab0w 4374* The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ab0 4375 using implicit substitution, which requires fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       ({𝑥𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓)
 
Theoremab0 4375 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4381 (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 2942). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2811, ax-8 2109. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremab0OLD 4376 Obsolete version of ab0 4375 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2811, ax-8 2109. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremab0ALT 4377 Alternate proof of ab0 4375, shorter but using more axioms. (Contributed by BJ, 19-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremdfnf5 4378 Characterization of nonfreeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))
 
Theoremab0orv 4379* 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 Gino Giotto, 30-Aug-2024.)
({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
 
Theoremab0orvALT 4380* Alternate proof of ab0orv 4379, 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.)
({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅)
 
Theoremabn0 4381 Nonempty class abstraction. See also ab0 4375. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2811, ax-8 2109. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
Theoremabn0OLD 4382 Obsolete version of abn0 4381 as of 30-Aug-2024. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
Theoremrab0 4383 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} = ∅
 
Theoremrabeq0w 4384* Condition for a restricted class abstraction to be empty. Version of rabeq0 4385 using implicit substitution, which does not require ax-10 2138, ax-11 2155, ax-12 2172, but requires ax-8 2109. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑦𝐴 ¬ 𝜓)
 
Theoremrabeq0 4385 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 4386 Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
 
Theoremrabxm 4387* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
 
Theoremrabnc 4388 Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
 
Theoremelneldisj 4389* 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 4390* 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 4391 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 4392 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 4393 The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∪ 𝐴) = 𝐴
 
Theorem0in 4394 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∩ 𝐴) = ∅
 
Theoreminv1 4395 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 4396 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 4397 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 4398 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 4399 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)
 
Theoremsseq0 4400 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
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