P. 42 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	sspsstr 4101	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psssstr 4102	Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
Theorem	psstrd 4103	Proper subclass inclusion is transitive. Deduction form of psstr 4100. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	sspsstrd 4104	Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4101. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊊ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	psssstrd 4105	Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4102. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊊ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐶)
Theorem	npss 4106	A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3992. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (¬ 𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 → 𝐴 = 𝐵))
Theorem	ssnelpss 4107	A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
⊢ (𝐴 ⊆ 𝐵 → ((𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ 𝐴) → 𝐴 ⊊ 𝐵))
Theorem	ssnelpssd 4108	Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 4107. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
Theorem	ssexnelpss 4109*	If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝑥 ∉ 𝐴) → 𝐴 ⊊ 𝐵)
2.1.13  The difference, union, and intersection of two classes
2.1.13.1  The difference of two classes
Theorem	dfdif3 4110*	Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)
⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
Theorem	difeq1 4111	Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
Theorem	difeq2 4112	Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 = 𝐵 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
Theorem	difeq12 4113	Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeq1i 4114	Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)
Theorem	difeq2i 4115	Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵)
Theorem	difeq12i 4116	Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)
Theorem	difeq1d 4117	Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶))
Theorem	difeq2d 4118	Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵))
Theorem	difeq12d 4119	Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷))
Theorem	difeqri 4120*	Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∖ 𝐵) = 𝐶
Theorem	nfdif 4121	Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2129, ax-11 2146, ax-12 2166. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
Theorem	nfdifOLD 4122	Obsolete version of nfdif 4121 as of 14-May-2025. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)
Theorem	eldifi 4123	Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	eldifn 4124	Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶)
Theorem	elndif 4125	A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵))
Theorem	neldif 4126	Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ 𝐶)) → 𝐴 ∈ 𝐶)
Theorem	difdif 4127	Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
Theorem	difss 4128	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
Theorem	difssd 4129	A difference of two classes is contained in the minuend. Deduction form of difss 4128. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
Theorem	difss2 4130	If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)
Theorem	difss2d 4131	If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4130. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ssdifss 4132	Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ddif 4133	Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
⊢ (V ∖ (V ∖ 𝐴)) = 𝐴
Theorem	ssconb 4134	Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))
Theorem	sscon 4135	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdif 4136	Difference law for subsets. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	ssdifd 4137	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4136. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	sscond 4138	If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4135. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdifssd 4139	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 4132. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ssdif2d 4140	If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶))
Theorem	raldifb 4141	Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
Theorem	rexdifi 4142	Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
Theorem	complss 4143	Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.)
⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
Theorem	compleq 4144	Two classes are equal if and only if their complements are equal. (Contributed by BJ, 19-Mar-2021.)
⊢ (𝐴 = 𝐵 ↔ (V ∖ 𝐴) = (V ∖ 𝐵))
2.1.13.2  The union of two classes
Theorem	elun 4145	Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
Theorem	elunnel1 4146	A member of a union that is not a member of the first class, is a member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶)
Theorem	elunnel2 4147	A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	uneqri 4148*	Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∪ 𝐵) = 𝐶
Theorem	unidm 4149	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ∪ 𝐴) = 𝐴
Theorem	uncom 4150	Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
Theorem	equncom 4151	If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4151 was automatically derived from equncomVD 44449 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵))
Theorem	equncomi 4152	Inference form of equncom 4151. equncomi 4152 was automatically derived from equncomiVD 44450 using the tools program translate_without_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
⊢ 𝐴 = (𝐵 ∪ 𝐶)    ⇒   ⊢ 𝐴 = (𝐶 ∪ 𝐵)
Theorem	uneq1 4153	Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
Theorem	uneq2 4154	Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
Theorem	uneq12 4155	Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
Theorem	uneq1i 4156	Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)
Theorem	uneq2i 4157	Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)
Theorem	uneq12i 4158	Equality inference for the union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)
Theorem	uneq1d 4159	Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
Theorem	uneq2d 4160	Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
Theorem	uneq12d 4161	Equality deduction for the union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
Theorem	nfun 4162	Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2129, ax-11 2146, ax-12 2166. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)
Theorem	nfunOLD 4163	Obsolete version of nfun 4162 as of 14-May-2025. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)
Theorem	unass 4164	Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶))
Theorem	un12 4165	A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶))
Theorem	un23 4166	A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵)
Theorem	un4 4167	A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷))
Theorem	unundi 4168	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))
Theorem	unundir 4169	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))
Theorem	ssun1 4170	Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
Theorem	ssun2 4171	Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
Theorem	ssun3 4172	Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
Theorem	ssun4 4173	Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))
Theorem	elun1 4174	Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶))
Theorem	elun2 4175	Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵))
Theorem	elunant 4176	A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑)))
Theorem	unss1 4177	Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
Theorem	ssequn1 4178	A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
Theorem	unss2 4179	Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵))
Theorem	unss12 4180	Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
Theorem	ssequn2 4181	A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)
Theorem	unss 4182	The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶)
Theorem	unssi 4183	An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ 𝐴 ⊆ 𝐶    &   ⊢ 𝐵 ⊆ 𝐶    ⇒   ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶
Theorem	unssd 4184	A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)
Theorem	unssad 4185	If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4182. Partial converse of unssd 4184. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	unssbd 4186	If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4182. Partial converse of unssd 4184. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	ssun 4187	A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))
Theorem	rexun 4188	Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ralunb 4189	Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	ralun 4190	Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑)
2.1.13.3  The intersection of two classes
Theorem	elini 4191	Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)
Theorem	elind 4192	Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
Theorem	elinel1 4193	Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	elinel2 4194	Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶)
Theorem	elin2 4195	Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑋 = (𝐵 ∩ 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
Theorem	elin1d 4196	Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐴)
Theorem	elin2d 4197	Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	elin3 4198	Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷)    ⇒   ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))
Theorem	incom 4199	Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) (Proof shortened by SN, 12-Dec-2023.)
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
Theorem	ineqcom 4200	Two ways of expressing that two classes have a given intersection. This is often used when that given intersection is the empty set, in which case the statement displays two ways of expressing that two classes are disjoint (when 𝐶 = ∅: ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅)). (Contributed by Peter Mazsa, 22-Mar-2017.)
⊢ ((𝐴 ∩ 𝐵) = 𝐶 ↔ (𝐵 ∩ 𝐴) = 𝐶)