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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iinxprg 4601* | 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 4602* | 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 4603* | 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 | iunun 4604 | 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 4605 | 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 | iunxdif3 4606* | An indexed union where some terms are the empty set. See iunxdif2 4568. (Contributed by Thierry Arnoux, 4-May-2020.) |
⊢ Ⅎ𝑥𝐸 ⇒ ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunxprg 4607* | 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 4608* | 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 4609* | 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 4610* | 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 4611 | Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | ||
Theorem | pwssb 4612* | Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | ||
Theorem | elpwpw 4613 | 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 4614* | 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 4615* | 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 4616 | Relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) | ||
Theorem | iinpw 4617* | 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 4618* | 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 | rintn0 4619 | 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 4620 | 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 4621* | 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 4622* | Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.) |
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | ||
Theorem | disjss2 4623 | 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 4624 | 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 4625* | 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 4626* | 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 4627* | Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) | ||
Theorem | disjeq1d 4628* | 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 4629* | 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 4630* | 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 4631* | 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 | nfdisj 4632 | Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfdisj1 4633 | Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | disjor 4634* | 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 4635* | 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 4636* | 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 4637* | 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 4638* | 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 4639* | 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 4640* | 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 4641* | 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 4642* | 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 4643* | 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 4644 | Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ 𝐴 {𝑥} | ||
Theorem | 0disj 4645 | Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ 𝐴 ∅ | ||
Theorem | disjxsn 4646* | A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ {𝐴}𝐵 | ||
Theorem | disjx0 4647 | An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ Disj 𝑥 ∈ ∅ 𝐵 | ||
Theorem | disjprg 4648* | 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 4649* | 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 | disjxiunOLD 4650* | Obsolete proof of disjxiun 4649 as of 27-May-2021. 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.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (Disj 𝑦 ∈ 𝐴 𝐵 → (Disj 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ (∀𝑦 ∈ 𝐴 Disj 𝑥 ∈ 𝐵 𝐶 ∧ Disj 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶))) | ||
Theorem | disjxun 4651* | 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 4652* | 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 4653 | 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 10267.) |
wff 𝐴𝑅𝐵 | ||
Definition | df-br 4654 | 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 10079 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {〈2, 6〉, 〈3, 9〉} → 3𝑅9) (ex-br 27288). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5121, and in particular 𝑅 may be a function (see df-fun 5890). 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 7101). (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | ||
Theorem | breq 4655 | Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.) |
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) | ||
Theorem | breq1 4656 | Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | breq2 4657 | Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.) |
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | ||
Theorem | breq12 4658 | Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | breqi 4659 | Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) | ||
Theorem | breq1i 4660 | Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) | ||
Theorem | breq2i 4661 | Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) | ||
Theorem | breq12i 4662 | 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 4663 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) | ||
Theorem | breqd 4664 | Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷)) | ||
Theorem | breq2d 4665 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) | ||
Theorem | breq12d 4666 | 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 4667 | 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 4668 | Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶𝐴𝐷) ⇒ ⊢ (𝜑 → 𝐶𝐵𝐷) | ||
Theorem | breqan12d 4669 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | breqan12rd 4670 | Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) | ||
Theorem | eqnbrtrd 4671 | 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 4672 | 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 4673 | 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 4674 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | eqbrtrd 4675 | Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | eqbrtrri 4676 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐴𝑅𝐶 ⇒ ⊢ 𝐵𝑅𝐶 | ||
Theorem | eqbrtrrd 4677 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐶) | ||
Theorem | breqtri 4678 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐵 = 𝐶 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | breqtrd 4679 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | breqtrri 4680 | Substitution of equal classes into a binary relation. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐶 = 𝐵 ⇒ ⊢ 𝐴𝑅𝐶 | ||
Theorem | breqtrrd 4681 | Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | 3brtr3i 4682 | Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐴 = 𝐶 & ⊢ 𝐵 = 𝐷 ⇒ ⊢ 𝐶𝑅𝐷 | ||
Theorem | 3brtr4i 4683 | Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ 𝐶 = 𝐴 & ⊢ 𝐷 = 𝐵 ⇒ ⊢ 𝐶𝑅𝐷 | ||
Theorem | 3brtr3d 4684 | 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 4685 | 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 4686 | 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 4687 | 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 | syl5eqbr 4688 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ 𝐴 = 𝐵 & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl5eqbrr 4689 | A chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.) |
⊢ 𝐵 = 𝐴 & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl5breq 4690 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ 𝐴𝑅𝐵 & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl5breqr 4691 | A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
⊢ 𝐴𝑅𝐵 & ⊢ (𝜑 → 𝐶 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl6eqbr 4692 | A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl6eqbrr 4693 | A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.) |
⊢ (𝜑 → 𝐵 = 𝐴) & ⊢ 𝐵𝑅𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl6breq 4694 | A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ 𝐵 = 𝐶 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | syl6breqr 4695 | A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.) |
⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ 𝐶 = 𝐵 ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ssbrd 4696 | Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) | ||
Theorem | ssbri 4697 | 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 4698 | Deduction version of bound-variable hypothesis builder nfbr 4699. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝑅) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵) | ||
Theorem | nfbr 4699 | 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 4700* | Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.) |
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴}) |
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