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Theorem List for Metamath Proof Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfsymdif2 4201* Alternate definition of the symmetric difference. (Contributed by BJ, 30-Apr-2020.)
(𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}

Theoremsymdifass 4202 Symmetric difference is associative. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by BJ, 7-Sep-2022.)
((𝐴𝐵) △ 𝐶) = (𝐴 △ (𝐵𝐶))

Theoremdifsssymdif 4203 The symmetric difference contains one of the differences. (Proposed by BJ, 18-Aug-2022.) (Contributed by AV, 19-Aug-2022.)
(𝐴𝐵) ⊆ (𝐴𝐵)

Theoremdifsymssdifssd 4204 If the symmetric difference is contained in 𝐶, so is one of the differences. (Contributed by AV, 17-Aug-2022.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

2.1.13.5  Combinations of difference, union, and intersection of two classes

Theoremunabs 4205 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminabs 4206 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∩ (𝐴𝐵)) = 𝐴

Theoremnssinpss 4207 Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
𝐴𝐵 ↔ (𝐴𝐵) ⊊ 𝐴)

Theoremnsspssun 4208 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
𝐴𝐵𝐵 ⊊ (𝐴𝐵))

Theoremdfss4 4209 Subclass defined in terms of class difference. See comments under dfun2 4210. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴)

Theoremdfun2 4210 An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4211 and dfss4 4209 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation (class difference). (Contributed by NM, 10-Jun-2004.)
(𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Theoremdfin2 4211 An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4210. Another version is given by dfin4 4218. (Contributed by NM, 10-Jun-2004.)
(𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Theoremdifin 4212 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)

Theoremssdifim 4213 Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.)
((𝐴𝑉𝐵 = (𝑉𝐴)) → 𝐴 = (𝑉𝐵))

Theoremssdifsym 4214 Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.)
((𝐴𝑉𝐵𝑉) → (𝐵 = (𝑉𝐴) ↔ 𝐴 = (𝑉𝐵)))

Theoremdfss5 4215* 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 4216 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 4217 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 4218 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 4219 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)

Theoremindif 4220 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoremindif2 4221 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremindif1 4222 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝐶) ∩ 𝐵) = ((𝐴𝐵) ∖ 𝐶)

Theoremindifcom 4223 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Theoremindi 4224 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 4225 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 4226 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremundir 4227 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremunineq 4228 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 4229 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 4230 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremdifundir 4231 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremdifindi 4232 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremdifindir 4233 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremindifdir 4234 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremdifdif2 4235 Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremundm 4236 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 4237 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 4238 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremundif3 4239 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 4240 Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))

Theoremdif32 4241 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)

Theoremdifabs 4242 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremdfsymdif3 4243 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

Theoremunab 4244 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoreminab 4245 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoremdifab 4246 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∖ {𝑥𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}

Theoremnotab 4247 A class abstraction defined by a negation. (Contributed by FL, 18-Sep-2010.)
{𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})

Theoremunrab 4248 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab 4249 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab2 4250* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Theoremdifrab 4251 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Theoremdfrab3 4252* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})

Theoremdfrab2 4253* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
{𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)

Theoremnotrab 4254* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}

Theoremdfrab3ss 4255* 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 4256 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

Theoremreuss2 4257* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuss 4258* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)

Theoremreuun1 4259* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuun2 4260* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))

Theoremreupick 4261* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))

Theoremreupick3 4262* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))

Theoremreupick2 4263* 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 4264* 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 4265 Extend class notation to include the empty set.
class

Definitiondf-nul 4266 Define the empty set. More precisely, we should write "empty class". It will be posited in ax-nul 5186 that an empty set exists. Then, by uniqueness among classes (eq0 4280, as opposed to the weaker uniqueness among sets, nulmo 2799), it will follow that is indeed a set (0ex 5187). Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 4267. (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)

Theoremdfnul2 4267 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2145, ax-11 2161, and ax-12 2178. (Revised by Steven Nguyen, 3-May-2023.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Theoremdfnul3 4268 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

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 2145, ax-11 2161, and ax-12 2178. (Revised by Steven Nguyen, 3-May-2023.)
¬ 𝐴 ∈ ∅

Theoremnel02 4270 The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.)
(𝐴 = ∅ → ¬ 𝐵𝐴)

Theoremn0i 4271 If a class has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)

Theoremne0i 4272 If a class has elements, then it is nonempty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)

Theoremne0d 4273 Deduction form of ne0i 4272. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐴)       (𝜑𝐴 ≠ ∅)

Theoremn0ii 4274 If a class has elements, then it is not empty. Inference associated with n0i 4271. (Contributed by BJ, 15-Jul-2021.)
𝐴𝐵        ¬ 𝐵 = ∅

Theoremne0ii 4275 If a class has elements, then it is nonempty. Inference associated with ne0i 4272. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴𝐵       𝐵 ≠ ∅

Theoremvn0 4276 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅

Theoremeq0f 4277 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 4278 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 4281 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 4279 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 4282 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 4280* 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 4281* 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 4282* 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 4283* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅

Theoremreximdva0 4284* Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)

Theoremrspn0 4285* Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))

Theoremn0rex 4286* 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 4287* 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 4288* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))

Theoremrex0 4289 Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑

Theoremreu0 4290 Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
¬ ∃!𝑥 ∈ ∅ 𝜑

Theoremrmo0 4291 Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
∃*𝑥 ∈ ∅ 𝜑

Theorem0el 4292* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)

Theoremn0el 4293* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)

Theoremeqeuel 4294* 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 4295 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = ∅)

Theoremdifn0 4296 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 4297 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)

Theorempssdif 4298 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)

Theoremndisj 4299* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))

Theoremdifin0ss 4300 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
(((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))

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