P. 52 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iunxiun 5101*	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	iinuni 5102*	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ (𝐴 ∪ ∩ 𝐵) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥)
Theorem	iununi 5103*	A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))
Theorem	sspwuni 5104	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
Theorem	pwssb 5105*	Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
Theorem	elpwpw 5106	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 5107*	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 5108*	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 5109	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵))
Theorem	iinpw 5110*	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 5111*	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	intss2 5112	A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.)
⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋))
Theorem	rintn0 5113	Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
2.1.22  Disjointness
Syntax	wdisj 5114	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 5115*	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 5116*	Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
Theorem	disjss2 5117	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 5118	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 5119*	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 5120*	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 5121*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
Theorem	disjeq1d 5122*	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 5123*	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 5124*	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 5125*	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	nfdisjw 5126*	Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5127 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵
Theorem	nfdisj 5127	Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfdisjw 5126 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵
Theorem	nfdisj1 5128	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵
Theorem	disjor 5129*	Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))
Theorem	disjors 5130*	Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
Theorem	disji2 5131*	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	disji 5132*	Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)    ⇒   ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐶 ∧ 𝑍 ∈ 𝐷)) → 𝑋 = 𝑌)
Theorem	invdisj 5133*	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	invdisjrabw 5134*	Version of invdisjrab 5135 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Gino Giotto, 26-Jan-2024.)
⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}
Theorem	invdisjrab 5135*	The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.)
⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}
Theorem	disjiun 5136*	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	disjord 5137*	Conditions for a collection of sets 𝐴(𝑎) for 𝑎 ∈ 𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.)
⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑎 = 𝑏)    ⇒   ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴)
Theorem	disjiunb 5138*	Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖 ∈ 𝐴 and 𝑥 ∈ 𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.)
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐷)    &   ⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸)    ⇒   ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅))
Theorem	disjiund 5139*	Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎 ∈ 𝑉 and 𝑏 ∈ 𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.)
⊢ (𝑎 = 𝑐 → 𝐴 = 𝐶)    &   ⊢ (𝑏 = 𝑑 → 𝐶 = 𝐷)    &   ⊢ (𝑎 = 𝑐 → 𝑊 = 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐷) → 𝑎 = 𝑐)    ⇒   ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ 𝑊 𝐴)
Theorem	sndisj 5140	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
Theorem	0disj 5141	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ 𝐴 ∅
Theorem	disjxsn 5142*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ {𝐴}𝐵
Theorem	disjx0 5143	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ ∅ 𝐵
Theorem	disjprgw 5144*	Version of disjprg 5145 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Gino Giotto, 26-Jan-2024.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))
Theorem	disjprg 5145*	A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))
Theorem	disjxiun 5146*	An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that 𝐵(𝑦) and 𝐶(𝑥) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by JJ, 27-May-2021.)
⊢ (Disj 𝑦 ∈ 𝐴 𝐵 → (Disj 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ (∀𝑦 ∈ 𝐴 Disj 𝑥 ∈ 𝐵 𝐶 ∧ Disj 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶)))
Theorem	disjxun 5147*	The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 ∩ 𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
Theorem	disjss3 5148*	Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅) → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
2.1.23  Binary relations
Syntax	wbr 5149	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. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 11445.)
wff 𝐴𝑅𝐵
Definition	df-br 5150	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. Class 𝑅 often denotes a relation such as "< " that compares two classes 𝐴 and 𝐵, which might be numbers such as 1 and 2 (see df-ltxr 11253 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) (ex-br 29684). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5684, and in particular 𝑅 may be a function (see df-fun 6546). 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 7904). (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	breq 5151	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
Theorem	breq1 5152	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
Theorem	breq2 5153	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
Theorem	breq12 5154	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breqi 5155	Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)
Theorem	breq1i 5156	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)
Theorem	breq2i 5157	Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)
Theorem	breq12i 5158	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 5159	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
Theorem	breqd 5160	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷))
Theorem	breq2d 5161	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
Theorem	breq12d 5162	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 5163	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 5164	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶𝐴𝐷)    ⇒   ⊢ (𝜑 → 𝐶𝐵𝐷)
Theorem	breqan12d 5165	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breqan12rd 5166	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	eqnbrtrd 5167	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 5168	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 5169	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 5170	Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	eqbrtrd 5171	Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrri 5172	Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴𝑅𝐶    ⇒   ⊢ 𝐵𝑅𝐶
Theorem	eqbrtrrd 5173	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐶)
Theorem	breqtri 5174	Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	breqtrd 5175	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrri 5176	Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	breqtrrd 5177	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	3brtr3i 5178	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶𝑅𝐷
Theorem	3brtr4i 5179	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶𝑅𝐷
Theorem	3brtr3d 5180	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 5181	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 5182	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 5183	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 5184	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrrid 5185	A chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrid 5186	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrrid 5187	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
⊢ 𝐴𝑅𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrdi 5188	A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrrdi 5189	A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrdi 5190	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrrdi 5191	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ssbrd 5192	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbr 5193	Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbri 5194	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 5195	Deduction version of bound-variable hypothesis builder nfbr 5196. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝑅)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Theorem	nfbr 5196	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 5197*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
Theorem	br0 5198	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
⊢ ¬ 𝐴∅𝐵
Theorem	brne0 5199	If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)
Theorem	brun 5200	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))