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	elin1d 4101	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 4102	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 4103	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 4104	Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
Theorem	nel2nelin 4105	Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶))
Theorem	incom 4106	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 4107	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 4108	Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4107. 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 4109*	Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∩ 𝐵) = 𝐶
Theorem	ineq1 4110	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 4111	Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
Theorem	ineq12 4112	Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷))
Theorem	ineq1i 4113	Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)
Theorem	ineq2i 4114	Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)
Theorem	ineq12i 4115	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 4116	Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))
Theorem	ineq2d 4117	Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵))
Theorem	ineq12d 4118	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 4119	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 4120	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 4121	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 2177, ax-11 2193, ax-12 2212. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵)
Theorem	rabbi2dva 4122*	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 4123	Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∩ 𝐴) = 𝐴
Theorem	inass 4124	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 4125	A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶))
Theorem	in32 4126	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 4127	A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴))
Theorem	in31 4128	A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴)
Theorem	inrot 4129	Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵)
Theorem	in4 4130	Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷))
Theorem	inindi 4131	Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶))
Theorem	inindir 4132	Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶))
Theorem	inss1 4133	The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
Theorem	inss2 4134	The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
Theorem	ssin 4135	Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))
Theorem	ssini 4136	An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
⊢ 𝐴 ⊆ 𝐵    &   ⊢ 𝐴 ⊆ 𝐶    ⇒   ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶)
Theorem	ssind 4137	A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶))
Theorem	ssrin 4138	Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
Theorem	sslin 4139	Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵))
Theorem	ssrind 4140	Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶))
Theorem	ss2in 4141	Intersection of subclasses. (Contributed by NM, 5-May-2000.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷))
Theorem	ssinss1 4142	Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	ssinss1OLD 4143	Obsolete version of ssinss1 4142 as of 10-Jun-2026. (Contributed by NM, 14-Sep-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	ssinss1d 4144	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	inss 4145	Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	ralin 4146	Restricted universal quantification over intersection. (Contributed by Peter Mazsa, 8-Sep-2023.)
⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑))
Theorem	rexin 4147	Restricted existential quantification over intersection. (Contributed by Peter Mazsa, 17-Dec-2018.)
⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐵)𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝜑))
Theorem	dfss7 4148*	Alternate definition of subclass relationship. (Contributed by AV, 1-Aug-2022.)
⊢ (𝐵 ⊆ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = 𝐵)
2.1.12.3  The union of two classes
Syntax	cun 4149	Syntax for the union of two classes.
class (𝐴 ∪ 𝐵)
Theorem	unjust 4150*	Soundness justification theorem for df-un 4151. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
Definition	df-un 4151*	Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 31031). Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 4047) and intersection (𝐴 ∩ 𝐵) (df-in 4091). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 4215. For union defined in terms of intersection, see dfun3 4221. (Contributed by NM, 23-Aug-1993.)
⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
Theorem	elun 4152	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 4153	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 4154	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 4155*	Inference from membership to union. (Contributed by NM, 21-Jun-1993.)
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)    ⇒   ⊢ (𝐴 ∪ 𝐵) = 𝐶
Theorem	unidm 4156	Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 21-Jun-1993.)
⊢ (𝐴 ∪ 𝐴) = 𝐴
Theorem	uncom 4157	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 4158	If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 4158 was automatically derived from equncomVD 45944 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 4159	Inference form of equncom 4158. equncomi 4159 was automatically derived from equncomiVD 45945 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 4160	Equality theorem for the union of two classes. (Contributed by NM, 15-Jul-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶))
Theorem	uneq2 4161	Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵))
Theorem	uneq12 4162	Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
Theorem	uneq1i 4163	Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)
Theorem	uneq2i 4164	Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)
Theorem	uneq12i 4165	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 4166	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 4167	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 4168	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 4169	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 2177, ax-11 2193, ax-12 2212. (Revised by SN, 14-May-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)
Theorem	unass 4170	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 4171	A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶))
Theorem	un23 4172	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 4173	A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷))
Theorem	unundi 4174	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))
Theorem	unundir 4175	Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))
Theorem	ssun1 4176	Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
Theorem	ssun2 4177	Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴)
Theorem	ssun3 4178	Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
Theorem	ssun4 4179	Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))
Theorem	elun1 4180	Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶))
Theorem	elun2 4181	Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵))
Theorem	elunant 4182	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 4183	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 4184	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 4185	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 4186	Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
Theorem	ssequn2 4187	A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵)
Theorem	unss 4188	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 4189	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 4190	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 4191	If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4188. Partial converse of unssd 4190. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
Theorem	unssbd 4192	If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4188. Partial converse of unssd 4190. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
Theorem	ssun 4193	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 4194	Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑))
Theorem	ralunb 4195	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 4196	Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑)
2.1.12.4  The symmetric difference of two classes
Syntax	csymdif 4197	Declare the syntax for symmetric difference.
class (𝐴 △ 𝐵)
Definition	df-symdif 4198	Define the symmetric difference of two classes. Alternate definitions are dfsymdif2 4206, dfsymdif3 4251 and dfsymdif4 4204. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴))
Theorem	symdifcom 4199	Symmetric difference is commutative. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 △ 𝐵) = (𝐵 △ 𝐴)
Theorem	symdifeq1 4200	Equality theorem for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 = 𝐵 → (𝐴 △ 𝐶) = (𝐵 △ 𝐶))