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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | disjss1 5001* | 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 5002* | Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) | ||
Theorem | disjeq1d 5003* | 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 5004* | 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 5005* | 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 5006* | 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 5007* | Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5008 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfdisj 5008 | Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfdisjw 5007 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfdisj1 5009 | Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | disjor 5010* | 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 5011* | 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 5012* | 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 5013* | 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 5014* | 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 5015* | Version of invdisjrab 5016 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} | ||
Theorem | invdisjrab 5016* | 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 5017* | 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 5018* | 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 5019* | 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 5020* | 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 5021 | Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | ||
Theorem | 0disj 5022 | Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ 𝐴 ∅ | ||
Theorem | disjxsn 5023* | A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ {𝐴}𝐵 | ||
Theorem | disjx0 5024 | An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ ∅ 𝐵 | ||
Theorem | disjprgw 5025* | Version of disjprg 5026 with a disjoint variable condition, which does not require ax-13 2379. (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 5026* | 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 5027* | 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 5028* | 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 5029* | 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)) | ||
Syntax | wbr 5030 | 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 | ||
Definition | df-br 5031 | 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 28216). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5526, and in particular 𝑅 may be a function (see df-fun 6326). 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 7600). (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | ||
Theorem | breq 5032 | Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.) |
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) | ||
Theorem | breq1 5033 | Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | breq2 5034 | Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | ||
Theorem | breq12 5035 | Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | breqi 5036 | Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) | ||
Theorem | breq1i 5037 | Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) | ||
Theorem | breq2i 5038 | Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) | ||
Theorem | breq12i 5039 | 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 5040 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | breqd 5041 | Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷)) | ||
Theorem | breq2d 5042 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | ||
Theorem | breq12d 5043 | 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 5044 | 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 5045 | Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶𝐴𝐷) ⇒ ⊢ (𝜑 → 𝐶𝐵𝐷) | ||
Theorem | breqan12d 5046 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | breqan12rd 5047 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | eqnbrtrd 5048 | 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 5049 | 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 5050 | 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 5051 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | eqbrtrd 5052 | Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrri 5053 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴𝑅𝐶 ⇒ ⊢ 𝐵𝑅𝐶 | ||
Theorem | eqbrtrrd 5054 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐶) | ||
Theorem | breqtri 5055 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | breqtrd 5056 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrri 5057 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | breqtrrd 5058 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | 3brtr3i 5059 | Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶𝑅𝐷 | ||
Theorem | 3brtr4i 5060 | Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶𝑅𝐷 | ||
Theorem | 3brtr3d 5061 | 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 5062 | 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 5063 | 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 5064 | 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 5065 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrrid 5066 | A chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrid 5067 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrrid 5068 | A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
⊢ 𝐴𝑅𝐵 & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrdi 5069 | A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrrdi 5070 | A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrdi 5071 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrrdi 5072 | A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ssbrd 5073 | Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) | ||
Theorem | ssbr 5074 | Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) | ||
Theorem | ssbri 5075 | 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 5076 | Deduction version of bound-variable hypothesis builder nfbr 5077. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝑅) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) | ||
Theorem | nfbr 5077 | 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 5078* | Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.) |
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) | ||
Theorem | br0 5079 | The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.) |
⊢ ¬ 𝐴∅𝐵 | ||
Theorem | brne0 5080 | 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 5081 | The union of two binary relations. (Contributed by NM, 21-Dec-2008.) |
⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵)) | ||
Theorem | brin 5082 | The intersection of two relations. (Contributed by FL, 7-Oct-2008.) |
⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) | ||
Theorem | brdif 5083 | The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.) |
⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) | ||
Theorem | sbcbr123 5084 | Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 22-Aug-2018.) |
⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | sbcbr 5085* | Move substitution in and out of a binary relation. (Contributed by NM, 23-Aug-2018.) |
⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵⦋𝐴 / 𝑥⦌𝑅𝐶) | ||
Theorem | sbcbr12g 5086* | Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | sbcbr1g 5087* | Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅𝐶)) | ||
Theorem | sbcbr2g 5088* | Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵𝑅⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | brsymdif 5089 | Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) | ||
Theorem | brralrspcev 5090* | Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.) |
⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 𝐴𝑅𝐵) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴𝑅𝑥) | ||
Theorem | brimralrspcev 5091* | Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.) |
⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓)) | ||
Syntax | copab 5092 | Extend class notation to include ordered-pair class abstraction (class builder). |
class {〈𝑥, 𝑦〉 ∣ 𝜑} | ||
Definition | df-opab 5093* | Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition does not require it (see dfid2 5428 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also called "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 7732. An example is given by ex-opab 28217. (Contributed by NM, 4-Jul-1994.) |
⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | ||
Theorem | opabss 5094* | The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} ⊆ 𝑅 | ||
Theorem | opabbid 5095 | Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) | ||
Theorem | opabbidv 5096* | Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} = {〈𝑥, 𝑦〉 ∣ 𝜒}) | ||
Theorem | opabbii 5097 | Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} | ||
Theorem | nfopab 5098* | Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜑} | ||
Theorem | nfopab1 5099 | The first abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | ||
Theorem | nfopab2 5100 | The second abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
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