P. 41 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	riin0 4001*	Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴)
Theorem	riinm 4002*	Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)
Theorem	iinxsng 4003*	A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Theorem	iinxprg 4004*	Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸))
Theorem	iunxsng 4005*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Theorem	iunxsn 4006*	A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶
Theorem	iunxsngf 4007*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Theorem	iunun 4008	Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	iunxun 4009	Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶)
Theorem	iunxprg 4010*	A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸))
Theorem	iunxiun 4011*	Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
⊢ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶
Theorem	iinuniss 4012*	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ (𝐴 ∪ ∩ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥)
Theorem	iununir 4013*	A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33 but with biconditional changed to implication. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
Theorem	sspwuni 4014	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
Theorem	pwssb 4015*	Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
Theorem	elpwpw 4016	Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
Theorem	pwpwab 4017*	The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)
⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
Theorem	pwpwssunieq 4018*	The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴
Theorem	elpwuni 4019	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵))
Theorem	iinpw 4020*	The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥
Theorem	iunpwss 4021*	Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴
Theorem	rintm 4022*	Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
2.1.21  Disjointness
Syntax	wdisj 4023	Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥 ∈ 𝐴, is a disjoint family.
wff Disj 𝑥 ∈ 𝐴 𝐵
Definition	df-disj 4024*	A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
Theorem	dfdisj2 4025*	Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
Theorem	disjss2 4026	If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵))
Theorem	disjeq2 4027	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶))
Theorem	disjeq2dv 4028*	Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶))
Theorem	disjss1 4029*	A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶))
Theorem	disjeq1 4030*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
Theorem	disjeq1d 4031*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
Theorem	disjeq12d 4032*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))
Theorem	cbvdisj 4033*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)
Theorem	cbvdisjv 4034*	Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶)
Theorem	nfdisjv 4035*	Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵
Theorem	nfdisj1 4036	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵
Theorem	disjnim 4037*	If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.)
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅))
Theorem	disjnims 4038*	If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
Theorem	disji2 4039*	Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋 ≠ 𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)    ⇒   ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑋 ≠ 𝑌) → (𝐶 ∩ 𝐷) = ∅)
Theorem	invdisj 4040*	If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦 ∈ 𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥 ∈ 𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵)
Theorem	invdisjrab 4041*	The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)
⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}
Theorem	disjiun 4042*	A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) = ∅)
Theorem	sndisj 4043	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
Theorem	0disj 4044	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ 𝐴 ∅
Theorem	disjxsn 4045*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ {𝐴}𝐵
Theorem	disjx0 4046	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ ∅ 𝐵
2.1.22  Binary relations
Syntax	wbr 4047	Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous.
wff 𝐴𝑅𝐵
Definition	df-br 4048	Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. This definition of relations is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 4952). (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	breq 4049	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
Theorem	breq1 4050	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
Theorem	breq2 4051	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
Theorem	breq12 4052	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breqi 4053	Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)
Theorem	breq1i 4054	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)
Theorem	breq2i 4055	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)
Theorem	breq12i 4056	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)
Theorem	breq1d 4057	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
Theorem	breqd 4058	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
Theorem	breq2d 4059	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
Theorem	breq12d 4060	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breq123d 4061	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))
Theorem	breqdi 4062	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶𝐴𝐷)    ⇒   ⊢ (𝜑 → 𝐶𝐵𝐷)
Theorem	breqan12d 4063	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breqan12rd 4064	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	eqnbrtrd 4065	Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)
Theorem	nbrne1 4066	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶)
Theorem	nbrne2 4067	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
⊢ ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴 ≠ 𝐵)
Theorem	eqbrtri 4068	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	eqbrtrd 4069	Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrri 4070	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴𝑅𝐶    ⇒   ⊢ 𝐵𝑅𝐶
Theorem	eqbrtrrd 4071	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐶)
Theorem	breqtri 4072	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	breqtrd 4073	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrri 4074	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	breqtrrd 4075	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	3brtr3i 4076	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶𝑅𝐷
Theorem	3brtr4i 4077	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶𝑅𝐷
Theorem	3brtr3d 4078	Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	3brtr4d 4079	Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	3brtr3g 4080	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	3brtr4g 4081	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	eqbrtrid 4082	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrrid 4083	B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrid 4084	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrrid 4085	B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
⊢ 𝐴𝑅𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrdi 4086	A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrrdi 4087	A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrdi 4088	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrrdi 4089	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ssbrd 4090	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbr 4091	Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbri 4092	Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)
Theorem	nfbrd 4093	Deduction version of bound-variable hypothesis builder nfbr 4094. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝑅)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Theorem	nfbr 4094	Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴𝑅𝐵
Theorem	brab1 4095*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
Theorem	br0 4096	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
⊢ ¬ 𝐴∅𝐵
Theorem	brne0 4097	If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4098. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)
Theorem	brm 4098*	If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)
Theorem	brun 4099	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))
Theorem	brin 4100	The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))