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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempwpwssunieq 5001* 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

Theoremelpwuni 5002 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))

Theoremiinpw 5003* 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 = 𝑥𝐴 𝒫 𝑥

Theoremiunpwss 5004* 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.)
𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴

Theoremintss2 5005 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 ≠ ∅ → 𝐴𝑋))

Theoremrintn0 5006 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)

2.1.22  Disjointness

Syntaxwdisj 5007 Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥𝐴, is a disjoint family.
wff Disj 𝑥𝐴 𝐵

Definitiondf-disj 5008* 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 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)

Theoremdfdisj2 5009* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))

Theoremdisjss2 5010 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.)
(∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))

Theoremdisjeq2 5011 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))

Theoremdisjeq2dv 5012* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))

Theoremdisjss1 5013* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))

Theoremdisjeq1 5014* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Theoremdisjeq1d 5015* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

Theoremdisjeq12d 5016* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))

Theoremcbvdisj 5017* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)

Theoremcbvdisjv 5018* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)

Theoremnfdisjw 5019* Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5020 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵

Theoremnfdisj 5020 Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker nfdisjw 5019 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵

Theoremnfdisj1 5021 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥Disj 𝑥𝐴 𝐵

Theoremdisjor 5022* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
(𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))

Theoremdisjors 5023* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))

Theoremdisji2 5024* Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)

Theoremdisji 5025* 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 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)

Theoreminvdisj 5026* If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)

Theoreminvdisjrabw 5027* Version of invdisjrab 5028 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Gino Giotto, 26-Jan-2024.)
Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}

Theoreminvdisjrab 5028* The restricted class abstractions {𝑥𝐵𝐶 = 𝑦} for distinct 𝑦𝐴 are disjoint. (Contributed by AV, 6-May-2020.)
Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}

Theoremdisjiun 5029* 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 𝑥𝐴 𝐵 ∧ (𝐶𝐴𝐷𝐴 ∧ (𝐶𝐷) = ∅)) → ( 𝑥𝐶 𝐵 𝑥𝐷 𝐵) = ∅)

Theoremdisjord 5030* Conditions for a collection of sets 𝐴(𝑎) for 𝑎𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.)
(𝑎 = 𝑏𝐴 = 𝐵)    &   ((𝜑𝑥𝐴𝑥𝐵) → 𝑎 = 𝑏)       (𝜑Disj 𝑎𝑉 𝐴)

Theoremdisjiunb 5031* Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖𝐴 and 𝑥𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.)
(𝑖 = 𝑗𝐵 = 𝐷)    &   (𝑖 = 𝑗𝐶 = 𝐸)       (Disj 𝑖𝐴 𝑥𝐵 𝐶 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ ( 𝑥𝐵 𝐶 𝑥𝐷 𝐸) = ∅))

Theoremdisjiund 5032* Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎𝑉 and 𝑏𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.)
(𝑎 = 𝑐𝐴 = 𝐶)    &   (𝑏 = 𝑑𝐶 = 𝐷)    &   (𝑎 = 𝑐𝑊 = 𝑋)    &   ((𝜑𝑥𝐴𝑥𝐷) → 𝑎 = 𝑐)       (𝜑Disj 𝑎𝑉 𝑏𝑊 𝐴)

Theoremsndisj 5033 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴 {𝑥}

Theorem0disj 5034 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴

Theoremdisjxsn 5035* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ {𝐴}𝐵

Theoremdisjx0 5036 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ ∅ 𝐵

Theoremdisjprgw 5037* Version of disjprg 5038 with a disjoint variable condition, which does not require ax-13 2391. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))

Theoremdisjprg 5038* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))

Theoremdisjxiun 5039* 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 𝑦𝐴 𝐵 → (Disj 𝑥 𝑦𝐴 𝐵𝐶 ↔ (∀𝑦𝐴 Disj 𝑥𝐵 𝐶Disj 𝑦𝐴 𝑥𝐵 𝐶)))

Theoremdisjxun 5040* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑦𝐶 = 𝐷)       ((𝐴𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴𝐵)𝐶 ↔ (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶 ∧ ∀𝑥𝐴𝑦𝐵 (𝐶𝐷) = ∅)))

Theoremdisjss3 5041* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))

2.1.23  Binary relations

Syntaxwbr 5042 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 10860.)
wff 𝐴𝑅𝐵

Definitiondf-br 5043 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 10669 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) (ex-br 28214). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5539, and in particular 𝑅 may be a function (see df-fun 6336). 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 7604). (Contributed by NM, 31-Dec-1993.)
(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)

Theorembreq 5044 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))

Theorembreq1 5045 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))

Theorembreq2 5046 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))

Theorembreq12 5047 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Theorembreqi 5048 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (𝐴𝑅𝐵𝐴𝑆𝐵)

Theorembreq1i 5049 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐴𝑅𝐶𝐵𝑅𝐶)

Theorembreq2i 5050 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐶𝑅𝐴𝐶𝑅𝐵)

Theorembreq12i 5051 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝑅𝐶𝐵𝑅𝐷)

Theorembreq1d 5052 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))

Theorembreqd 5053 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))

Theorembreq2d 5054 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))

Theorembreq12d 5055 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))

Theorembreq123d 5056 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝑅 = 𝑆)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))

Theorembreqdi 5057 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶𝐴𝐷)       (𝜑𝐶𝐵𝐷)

Theorembreqan12d 5058 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Theorembreqan12rd 5059 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))

Theoremeqnbrtrd 5060 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝑅𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)

Theoremnbrne1 5061 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)

Theoremnbrne2 5062 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)

Theoremeqbrtri 5063 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴 = 𝐵    &   𝐵𝑅𝐶       𝐴𝑅𝐶

Theoremeqbrtrd 5064 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theoremeqbrtrri 5065 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴 = 𝐵    &   𝐴𝑅𝐶       𝐵𝑅𝐶

Theoremeqbrtrrd 5066 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝑅𝐶)       (𝜑𝐵𝑅𝐶)

Theorembreqtri 5067 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴𝑅𝐵    &   𝐵 = 𝐶       𝐴𝑅𝐶

Theorembreqtrd 5068 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)

Theorembreqtrri 5069 Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐵       𝐴𝑅𝐶

Theorembreqtrrd 5070 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)

Theorem3brtr3i 5071 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝑅𝐷

Theorem3brtr4i 5072 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝑅𝐷

Theorem3brtr3d 5073 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝑅𝐷)

Theorem3brtr4d 5074 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝑅𝐷)

Theorem3brtr3g 5075 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝑅𝐷)

Theorem3brtr4g 5076 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝑅𝐷)

Theoremeqbrtrid 5077 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴 = 𝐵    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theoremeqbrtrrid 5078 A chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theorembreqtrid 5079 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴𝑅𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)

Theorembreqtrrid 5080 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
𝐴𝑅𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)

Theoremeqbrtrdi 5081 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)

Theoremeqbrtrrdi 5082 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)

Theorembreqtrdi 5083 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝑅𝐶)

Theorembreqtrrdi 5084 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝑅𝐶)

Theoremssbrd 5085 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))

Theoremssbr 5086 Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)
(𝐴𝐵 → (𝐶𝐴𝐷𝐶𝐵𝐷))

Theoremssbri 5087 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝐴𝐵       (𝐶𝐴𝐷𝐶𝐵𝐷)

Theoremnfbrd 5088 Deduction version of bound-variable hypothesis builder nfbr 5089. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝑅)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)

Theoremnfbr 5089 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝑅    &   𝑥𝐵       𝑥 𝐴𝑅𝐵

Theorembrab1 5090* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})

Theorembr0 5091 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
¬ 𝐴𝐵

Theorembrne0 5092 If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
(𝐴𝑅𝐵𝑅 ≠ ∅)

Theorembrun 5093 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))

Theorembrin 5094 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))

Theorembrdif 5095 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))

Theoremsbcbr123 5096 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 22-Aug-2018.)
([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)

Theoremsbcbr 5097* Move substitution in and out of a binary relation. (Contributed by NM, 23-Aug-2018.)
([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)

Theoremsbcbr12g 5098* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))

Theoremsbcbr1g 5099* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))

Theoremsbcbr2g 5100* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))

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