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Type | Label | Description |
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Statement | ||
Theorem | iununi 5101* | 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 5102 | Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | ||
Theorem | pwssb 5103* | Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | ||
Theorem | elpwpw 5104 | 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 5105* | 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 5106* | 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 5107 | Relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) | ||
Theorem | iinpw 5108* | 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 5109* | 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 5110 | 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 5111 | 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) | ||
Syntax | wdisj 5112 | 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 5113* | 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 5114* | Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.) |
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | ||
Theorem | disjss2 5115 | 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 5116 | 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 5117* | 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 5118* | 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 5119* | Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) | ||
Theorem | disjeq1d 5120* | 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 5121* | 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 5122* | 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 5123* | 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 5124* | Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5125 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2369. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfdisj 5125 | Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfdisjw 5124 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfdisj1 5126 | Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | disjor 5127* | 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 5128* | 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 5129* | 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 5130* | 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 5131* | 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 5132* | Version of invdisjrab 5133 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} | ||
Theorem | invdisjrab 5133* | 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 5134* | 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 5135* | 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 5136* | 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 5137* | 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 5138 | Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | ||
Theorem | 0disj 5139 | Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ 𝐴 ∅ | ||
Theorem | disjxsn 5140* | A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ {𝐴}𝐵 | ||
Theorem | disjx0 5141 | An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ ∅ 𝐵 | ||
Theorem | disjprgw 5142* | Version of disjprg 5143 with a disjoint variable condition, which does not require ax-13 2369. (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 5143* | 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 5144* | 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 5145* | 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 5146* | 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 5147 | 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 11449.) |
wff 𝐴𝑅𝐵 | ||
Definition | df-br 5148 | 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 11257 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) (ex-br 29951). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5682, and in particular 𝑅 may be a function (see df-fun 6544). 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 7906). (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) | ||
Theorem | breq 5149 | Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.) |
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) | ||
Theorem | breq1 5150 | Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | breq2 5151 | Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | ||
Theorem | breq12 5152 | Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | breqi 5153 | Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) | ||
Theorem | breq1i 5154 | Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) | ||
Theorem | breq2i 5155 | Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) | ||
Theorem | breq12i 5156 | 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 5157 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | breqd 5158 | Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷)) | ||
Theorem | breq2d 5159 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | ||
Theorem | breq12d 5160 | 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 5161 | 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 5162 | Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶𝐴𝐷) ⇒ ⊢ (𝜑 → 𝐶𝐵𝐷) | ||
Theorem | breqan12d 5163 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | breqan12rd 5164 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | eqnbrtrd 5165 | 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 5166 | 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 5167 | 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 5168 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | eqbrtrd 5169 | Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrri 5170 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴𝑅𝐶 ⇒ ⊢ 𝐵𝑅𝐶 | ||
Theorem | eqbrtrrd 5171 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐶) | ||
Theorem | breqtri 5172 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | breqtrd 5173 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrri 5174 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | breqtrrd 5175 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | 3brtr3i 5176 | Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶𝑅𝐷 | ||
Theorem | 3brtr4i 5177 | Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶𝑅𝐷 | ||
Theorem | 3brtr3d 5178 | 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 5179 | 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 5180 | 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 5181 | 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 5182 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrrid 5183 | A chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrid 5184 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrrid 5185 | A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
⊢ 𝐴𝑅𝐵 & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrdi 5186 | A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrrdi 5187 | A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrdi 5188 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrrdi 5189 | A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ssbrd 5190 | Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) | ||
Theorem | ssbr 5191 | Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) | ||
Theorem | ssbri 5192 | 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 5193 | Deduction version of bound-variable hypothesis builder nfbr 5194. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝑅) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) | ||
Theorem | nfbr 5194 | 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 5195* | Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.) |
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) | ||
Theorem | br0 5196 | The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.) |
⊢ ¬ 𝐴∅𝐵 | ||
Theorem | brne0 5197 | 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 5198 | The union of two binary relations. (Contributed by NM, 21-Dec-2008.) |
⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵)) | ||
Theorem | brin 5199 | The intersection of two relations. (Contributed by FL, 7-Oct-2008.) |
⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) | ||
Theorem | brdif 5200 | The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.) |
⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
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