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	difdif 4101	Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
Theorem	difss 4102	Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
Theorem	difssd 4103	A difference of two classes is contained in the minuend. Deduction form of difss 4102. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
Theorem	difss2 4104	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 4105	If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 4104. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ssdifss 4106	Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ddif 4107	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 4108	Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))
Theorem	sscon 4109	Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdif 4110	Difference law for subsets. (Contributed by NM, 28-May-1998.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	ssdifd 4111	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4110. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
Theorem	sscond 4112	If 𝐴 is contained in 𝐵, then (𝐶 ∖ 𝐵) is contained in (𝐶 ∖ 𝐴). Deduction form of sscon 4109. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴))
Theorem	ssdifssd 4113	If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is also contained in 𝐵. Deduction form of ssdifss 4106. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵)
Theorem	ssdif2d 4114	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 4115	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 4116	Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
Theorem	complss 4117	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 4118	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 4119	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 4120	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 4121	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 4122*	Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∪ 𝐵) = 𝐶
Theorem	unidm 4123	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ∪ 𝐴) = 𝐴
Theorem	uncom 4124	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 4125	If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4125 was automatically derived from equncomVD 44864 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 4126	Inference form of equncom 4125. equncomi 4126 was automatically derived from equncomiVD 44865 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 4127	Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
Theorem	uneq2 4128	Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
Theorem	uneq12 4129	Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
Theorem	uneq1i 4130	Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)
Theorem	uneq2i 4131	Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)
Theorem	uneq12i 4132	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 4133	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 4134	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 4135	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 4136	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 2142, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)
Theorem	nfunOLD 4137	Obsolete version of nfun 4136 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 4138	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 4139	A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶))
Theorem	un23 4140	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 4141	A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷))
Theorem	unundi 4142	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))
Theorem	unundir 4143	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))
Theorem	ssun1 4144	Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
Theorem	ssun2 4145	Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
Theorem	ssun3 4146	Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
Theorem	ssun4 4147	Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))
Theorem	elun1 4148	Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶))
Theorem	elun2 4149	Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵))
Theorem	elunant 4150	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 4151	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 4152	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 4153	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 4154	Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
Theorem	ssequn2 4155	A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)
Theorem	unss 4156	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 4157	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 4158	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 4159	If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4156. Partial converse of unssd 4158. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	unssbd 4160	If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4156. Partial converse of unssd 4158. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	ssun 4161	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 4162	Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ralunb 4163	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 4164	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 4165	Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)
Theorem	elind 4166	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 4167	Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵)
Theorem	elinel2 4168	Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶)
Theorem	elin2 4169	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 4170	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 4171	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 4172	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	nel1nelin 4173	Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
Theorem	nel2nelin 4174	Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
Theorem	incom 4175	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 4176	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)
Theorem	ineqcomi 4177	Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4176. Disjointness inference when 𝐶 = ∅. (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)
⊢ (𝐴 ∩ 𝐵) = 𝐶    ⇒   ⊢ (𝐵 ∩ 𝐴) = 𝐶
Theorem	ineqri 4178*	Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∩ 𝐵) = 𝐶
Theorem	ineq1 4179	Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.) (Proof shortened by SN, 20-Sep-2023.)
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
Theorem	ineq2 4180	Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
Theorem	ineq12 4181	Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
Theorem	ineq1i 4182	Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)
Theorem	ineq2i 4183	Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)
Theorem	ineq12i 4184	Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)
Theorem	ineq1d 4185	Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
Theorem	ineq2d 4186	Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
Theorem	ineq12d 4187	Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
Theorem	ineqan12d 4188	Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
Theorem	sseqin2 4189	A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
Theorem	nfin 4190	Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
Theorem	nfinOLD 4191	Obsolete version of nfin 4190 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	rabbi2dva 4192*	Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓))    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	inidm 4193	Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∩ 𝐴) = 𝐴
Theorem	inass 4194	Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))
Theorem	in12 4195	A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))
Theorem	in32 4196	A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵)
Theorem	in13 4197	A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))
Theorem	in31 4198	A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)
Theorem	inrot 4199	Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵)
Theorem	in4 4200	Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷))