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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremundi 4101 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 4102 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremundir 4103 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremunineq 4104 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 4105 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 4106 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremdifundir 4107 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremdifindi 4108 Distributive law for class difference. Theorem 40 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremdifindir 4109 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremindifdir 4110 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremdifdif2 4111 Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremundm 4112 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 4113 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 4114 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremundif3 4115 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 4116 Represent a class difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))

Theoremdif32 4117 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)

Theoremdifabs 4118 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremdfsymdif3 4119 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 4120 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoreminab 4121 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoremdifab 4122 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 4123 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
{𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})

Theoremunrab 4124 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab 4125 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab2 4126* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Theoremdifrab 4127 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Theoremdfrab3 4128* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})

Theoremdfrab2 4129* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
{𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)

Theoremnotrab 4130* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}

Theoremdfrab3ss 4131* 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 4132 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 4133* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuss 4134* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)

Theoremreuun1 4135* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuun2 4136* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))

Theoremreupick 4137* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))

Theoremreupick3 4138* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))

Theoremreupick2 4139* 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 4140* 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 4141 Extend class notation to include the empty set.
class

Definitiondf-nul 4142 Define the empty set. More precisely, we should write "empty class". It will be posited in ax-nul 5027 that an empty set exists. Then, by uniqueness among classes (eq0 4157, as opposed to the weaker uniqueness among sets, nulmo 2761), it will follow that is indeed a set (0ex 5028). Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 4143. (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 4143 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2135, ax-11 2150, and ax-12 2163. (Revised by Steven Nguyen, 3-May-2023.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Theoremdfnul2OLD 4144 Obsolete version of dfnul2 4143 as of 3-May-2023. Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) (New usage is discouraged.) (Proof modification is discouraged.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Theoremdfnul3 4145 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Theoremnoel 4146 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 2135, ax-11 2150, and ax-12 2163. (Revised by Steven Nguyen, 3-May-2023.)
¬ 𝐴 ∈ ∅

TheoremnoelOLD 4147 Obsolete version of noel 4146 as of 3-May-2023. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 𝐴 ∈ ∅

Theoremn0i 4148 If a class has elements, then it is not empty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)

Theoremne0i 4149 If a class has elements, then it is nonempty. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)

Theoremne0d 4150 Deduction form of ne0i 4149. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐵𝐴)       (𝜑𝐴 ≠ ∅)

Theoremn0ii 4151 If a class has elements, then it is not empty. Inference associated with n0i 4148. (Contributed by BJ, 15-Jul-2021.)
𝐴𝐵        ¬ 𝐵 = ∅

Theoremne0ii 4152 If a class has elements, then it is nonempty. Inference associated with ne0i 4149. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴𝐵       𝐵 ≠ ∅

Theoremvn0 4153 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅

Theoremeq0f 4154 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 4155 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 4158 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 4156 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 4159 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 4157* 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 4158* 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 4159* 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 4160* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅

Theoremreximdva0 4161* Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑𝐴 ≠ ∅) → ∃𝑥𝐴 𝜓)

Theoremrspn0 4162* Specialization for restricted generalization with a nonempty class. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 𝜑𝜑))

Theoremn0rex 4163* 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 4164* 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 4165* A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
(𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))

Theoremrex0 4166 Vacuous restricted existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑

Theoremreu0 4167 Vacuous restricted uniqueness is always false. (Contributed by AV, 3-Apr-2023.)
¬ ∃!𝑥 ∈ ∅ 𝜑

Theoremrmo0 4168 Vacuous restricted at-most-one quantifier is always true. (Contributed by AV, 3-Apr-2023.)
∃*𝑥 ∈ ∅ 𝜑

Theorem0el 4169* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)

Theoremn0el 4170* Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
(¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)

Theoremeqeuel 4171* 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 4172 Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵 ↔ (𝐴𝐵) = ∅)

Theoremdifn0 4173 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 4174 A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)

Theorempssdif 4175 A proper subclass has a nonempty difference. (Contributed by Mario Carneiro, 27-Apr-2016.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)

Theoremndisj 4176* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝑥𝐵))

Theoremdifin0ss 4177 Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
(((𝐴𝐵) ∩ 𝐶) = ∅ → (𝐶𝐴𝐶𝐵))

Theoreminssdif0 4178 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 4179 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 4180 Alternate proof of difid 4179. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅

Theoremdif0 4181 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 4182 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4185 (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 2970). (Contributed by BJ, 19-Mar-2021.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Theoremdfnf5 4183 Characterization of non-freeness in a formula in terms of its extension. (Contributed by BJ, 19-Mar-2021.)
(Ⅎ𝑥𝜑 ↔ ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))

Theoremab0orv 4184* 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 4185 Nonempty class abstraction. See also ab0 4182. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Theoremrab0 4186 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 4187 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 4188 Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.) (Revised by BJ, 16-Jul-2021.)
({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)

Theoremrabxm 4189* Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})

Theoremrabnc 4190* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅

Theoremelneldisj 4191* 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 4192* 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 4193 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 4194 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 ∩ ∅) = ∅

Theorem0in 4195 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∩ 𝐴) = ∅

Theoreminv1 4196 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 4197 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 4198 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 4199 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 4200 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)

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