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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssdifim 4201 Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.)
((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))
 
Theoremssdifsym 4202 Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)
((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))
 
Theoremdfss5 4203* Alternate definition of subclass relationship: a class 𝐴 is a subclass of another class 𝐵 iff each element of 𝐴 is equal to an element of 𝐵. (Contributed by AV, 13-Nov-2020.)
(𝐴𝐵 ↔ ∀𝑥𝐴𝑦𝐵 𝑥 = 𝑦)
 
Theoremdfun3 4204 Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
(𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
 
Theoremdfin3 4205 Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
(𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
 
Theoremdfin4 4206 Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
(𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
 
Theoreminvdif 4207 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
 
Theoremindif 4208 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 
Theoremindif2 4209 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
 
Theoremindif1 4210 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝐶) ∩ 𝐵) = ((𝐴𝐵) ∖ 𝐶)
 
Theoremindifcom 4211 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 
Theoremindi 4212 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 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremundi 4213 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 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoremindir 4214 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremundir 4215 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoremunineq 4216 Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)
(((𝐴𝐶) = (𝐵𝐶) ∧ (𝐴𝐶) = (𝐵𝐶)) ↔ 𝐴 = 𝐵)
 
Theoremuneqin 4217 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = (𝐴𝐵) ↔ 𝐴 = 𝐵)
 
Theoremdifundi 4218 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoremdifundir 4219 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremdifindi 4220 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremdifindir 4221 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoremindifdi 4222 Distribute intersection over difference. (Contributed by BTernaryTau, 14-Aug-2024.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐴𝐶))
 
Theoremindifdir 4223 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by BTernaryTau, 14-Aug-2024.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
 
TheoremindifdirOLD 4224 Obsolete version of indifdir 4223 as of 14-Aug-2024. (Contributed by Scott Fenton, 14-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
 
Theoremdifdif2 4225 Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremundm 4226 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
(V ∖ (𝐴𝐵)) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
 
Theoremindm 4227 De Morgan's law for intersection. Theorem 5.2(13') of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
(V ∖ (𝐴𝐵)) = ((V ∖ 𝐴) ∪ (V ∖ 𝐵))
 
Theoremdifun1 4228 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
 
Theoremundif3 4229 An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.) (Proof shortened by JJ, 13-Jul-2021.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ (𝐶𝐴))
 
Theoremdifin2 4230 Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))
 
Theoremdif32 4231 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)
 
Theoremdifabs 4232 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
 
Theoremsscon34b 4233 Relative complementation reverses inclusion of subclasses. Relativized version of complss 4085. (Contributed by RP, 3-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵 ↔ (𝐶𝐵) ⊆ (𝐶𝐴)))
 
Theoremrcompleq 4234 Two subclasses are equal if and only if their relative complements are equal. Relativized version of compleq 4086. (Contributed by RP, 10-Jun-2021.)
((𝐴𝐶𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐶𝐴) = (𝐶𝐵)))
 
Theoremdfsymdif3 4235 Alternate definition of the symmetric difference, given in Example 4.1 of [Stoll] p. 262 (the original definition corresponds to [Stoll] p. 13). (Contributed by NM, 17-Aug-2004.) (Revised by BJ, 30-Apr-2020.)
(𝐴𝐵) = ((𝐴𝐵) ∖ (𝐴𝐵))
 
2.1.13.6  Class abstractions with difference, union, and intersection of two classes
 
Theoremunabw 4236* Union of two class abstractions. Version of unab 4237 using implicit substitution, which does not require ax-8 2111, ax-10 2140, ax-12 2174. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))       ({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑦 ∣ (𝜒𝜃)}
 
Theoremunab 4237 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
 
Theoreminab 4238 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}
 
Theoremdifab 4239 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 4240 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 4241 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 4242* A class abstraction defined by a negation. Version of notab 4243 using implicit substitution, which does not require ax-10 2140, ax-12 2174. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦𝜓})
 
Theoremnotab 4243 A class abstraction defined by a negation. (Contributed by FL, 18-Sep-2010.)
{𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
 
Theoremunrab 4244 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
 
Theoreminrab 4245 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
 
Theoreminrab2 4246* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 
Theoremdifrab 4247 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}
 
Theoremdfrab3 4248* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
 
Theoremdfrab2 4249* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
{𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
 
Theoremnotrab 4250* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
 
Theoremdfrab3ss 4251* 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 4252 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 4253 Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))
 
Theoremreuss2 4254* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)
 
Theoremreuss 4255* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)
 
Theoremreuun1 4256* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)
 
Theoremreupick 4257* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
 
Theoremreupick3 4258* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
 
Theoremreupick2 4259* 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 4260* 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 4261 Extend class notation to include the empty set.
class
 
Definitiondf-nul 4262 Define the empty set. More precisely, we should write "empty class". It will be posited in ax-nul 5233 that an empty set exists. Then, by uniqueness among classes (eq0 4282, as opposed to the weaker uniqueness among sets, nulmo 2715), it will follow that is indeed a set (0ex 5234). Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 4264. (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 4263 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2111, df-clel 2817. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4264. (Revised by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ⊥}
 
Theoremdfnul2 4264 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2140, ax-11 2157, and ax-12 2174. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
Theoremdfnul3 4265 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
 
Theoremdfnul2OLD 4266 Obsolete version of dfnul2 4264 as of 23-Sep-2024. (Contributed by NM, 26-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
Theoremdfnul3OLD 4267 Obsolete version of dfnul4 4263 as of 23-Sep-2024. (Contributed by NM, 25-Mar-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
 
Theoremdfnul4OLD 4268 Obsolete version of dfnul4 4263 as of 23-Sep-2024. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = {𝑥 ∣ ⊥}
 
Theoremnoel 4269 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 2140, ax-11 2157, and ax-12 2174. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
¬ 𝐴 ∈ ∅
 
TheoremnoelOLD 4270 Obsolete version of noel 4269 as of 18-Sep-2024. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝐴 ∈ ∅
 
Theoremnel02 4271 The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
(𝐴 = ∅ → ¬ 𝐵𝐴)
 
Theoremn0i 4272 If a class has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)
 
Theoremne0i 4273 If a class has elements, then it is nonempty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)
 
Theoremne0d 4274 Deduction form of ne0i 4273. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐴)       (𝜑𝐴 ≠ ∅)
 
Theoremn0ii 4275 If a class has elements, then it is not empty. Inference associated with n0i 4272. (Contributed by BJ, 15-Jul-2021.)
𝐴𝐵        ¬ 𝐵 = ∅
 
Theoremne0ii 4276 If a class has elements, then it is nonempty. Inference associated with ne0i 4273. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴𝐵       𝐵 ≠ ∅
 
Theoremvn0 4277 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2111, df-clel 2817. (Revised by Gino Giotto, 6-Sep-2024.)
V ≠ ∅
 
Theoremvn0ALT 4278 Alternate proof of vn0 4277. Shorter, but requiring df-clel 2817, ax-8 2111. (Contributed by NM, 11-Sep-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
V ≠ ∅
 
Theoremeq0f 4279 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 4280 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 4284 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 4281 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 4285 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 4282* 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 2157, ax-12 2174. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2111, df-clel 2817. (Revised by Gino Giotto, 6-Sep-2024.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremeq0ALT 4283* Alternate proof of eq0 4282. Shorter, but requiring df-clel 2817, ax-8 2111. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2157, ax-12 2174. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremneq0 4284* 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 2157, ax-12 2174. (Revised by Gino Giotto, 28-Jun-2024.)
𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremn0 4285* 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 2157, ax-12 2174. (Revised by Gino Giotto, 28-Jun-2024.)
(𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremeq0OLDOLD 4286* Obsolete version of eq0 4282 as of 28-Jun-2024. (Contributed by NM, 29-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 
Theoremneq0OLD 4287* Obsolete version of neq0 4284 as of 28-Jun-2024. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremn0OLD 4288* Obsolete version of n0 4285 as of 28-Jun-2024. (Contributed by NM, 29-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
 
Theoremnel0 4289* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅
 
Theoremreximdva0 4290* Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)
 
Theoremrspn0 4291* Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.) Avoid ax-10 2140, ax-12 2174. (Revised by Gino Giotto, 28-Jun-2024.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))
 
Theoremrspn0OLD 4292* Obsolete version of rspn0 4291 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 4293* 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 4294* 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 4295* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
 
Theoremrex0 4296 Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑
 
Theoremreu0 4297 Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
¬ ∃!𝑥 ∈ ∅ 𝜑
 
Theoremrmo0 4298 Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
∃*𝑥 ∈ ∅ 𝜑
 
Theorem0el 4299* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
 
Theoremn0el 4300* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
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