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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | otthg 5401 | Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (〈𝐴, 𝐵, 𝐶〉 = 〈𝐷, 𝐸, 𝐹〉 ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | ||
Theorem | eqvinop 5402* | A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 = 〈𝐵, 𝐶〉 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 = 〈𝐵, 𝐶〉)) | ||
Theorem | sbcop1 5403* | The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of its first component. (Contributed by AV, 8-Apr-2023.) |
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑦〉 / 𝑧]𝜑) | ||
Theorem | sbcop 5404* | The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of each of its components. (Contributed by AV, 8-Apr-2023.) |
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑏〉 / 𝑧]𝜑) | ||
Theorem | copsexgw 5405* | Version of copsexg 5406 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | ||
Theorem | copsexg 5406* | Substitution of class 𝐴 for ordered pair 〈𝑥, 𝑦〉. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker copsexgw 5405 when possible. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 25-Aug-2019.) (New usage is discouraged.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | ||
Theorem | copsex2t 5407* | Closed theorem form of copsex2g 5408. (Contributed by NM, 17-Feb-2013.) |
⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | copsex2g 5408* | Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5410 to reduce axiom usage. (Revised by SN, 1-Sep-2024.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | copsex2gOLD 5409* | Obsolete version of copsex2g 5408 as of 1-Sep-2024. (Contributed by NM, 28-May-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | copsex4g 5410* | An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.) |
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 〈𝐶, 𝐷〉 = 〈𝑧, 𝑤〉) ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | 0nelop 5411 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opwo0id 5412 | An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.) |
⊢ 〈𝑋, 𝑌〉 = (〈𝑋, 𝑌〉 ∖ {∅}) | ||
Theorem | opeqex 5413 | Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) | ||
Theorem | oteqex2 5414 | Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.) |
⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → (𝐶 ∈ V ↔ 𝑇 ∈ V)) | ||
Theorem | oteqex 5415 | Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V))) | ||
Theorem | opcom 5416 | An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐵, 𝐴〉 ↔ 𝐴 = 𝐵) | ||
Theorem | moop2 5417* | "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∃*𝑥 𝐴 = 〈𝐵, 𝑥〉 | ||
Theorem | opeqsng 5418 | Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))) | ||
Theorem | opeqsn 5419 | Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})) | ||
Theorem | opeqpr 5420 | Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴}))) | ||
Theorem | snopeqop 5421 | Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({〈𝐴, 𝐵〉} = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴})) | ||
Theorem | propeqop 5422 | Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.) (Proof shortened by AV, 16-Jun-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐸 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} = 〈𝐸, 𝐹〉 ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵})))) | ||
Theorem | propssopi 5423 | If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 16-Jun-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐸 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ ({〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉} ⊆ 〈𝐸, 𝐹〉 → 𝐴 = 𝐶) | ||
Theorem | snopeqopsnid 5424 | Equivalence for an ordered pair of two identical singletons equal to a singleton of an ordered pair. (Contributed by AV, 24-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {〈𝐴, 𝐴〉} = 〈{𝐴}, {𝐴}〉 | ||
Theorem | mosubopt 5425* | "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.) |
⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑)) | ||
Theorem | mosubop 5426* | "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.) |
⊢ ∃*𝑥𝜑 ⇒ ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑) | ||
Theorem | euop2 5427* | Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃!𝑥∃𝑦(𝑥 = 〈𝐴, 𝑦〉 ∧ 𝜑) ↔ ∃!𝑦𝜑) | ||
Theorem | euotd 5428* | Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))) ⇒ ⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = 〈𝑎, 𝑏, 𝑐〉 ∧ 𝜓)) | ||
Theorem | opthwiener 5429 | Justification theorem for the ordered pair definition in Norbert Wiener, A simplification of the logic of relations, Proceedings of the Cambridge Philosophical Society, 1914, vol. 17, pp.387-390. It is also shown as a definition in [Enderton] p. 36 and as Exercise 4.8(b) of [Mendelson] p. 230. It is meaningful only for classes that exist as sets (i.e., are not proper classes). See df-op 4569 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | uniop 5430 | The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ 〈𝐴, 𝐵〉 = {𝐴, 𝐵} | ||
Theorem | uniopel 5431 | Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → ∪ 〈𝐴, 𝐵〉 ∈ ∪ 𝐶) | ||
Theorem | opthhausdorff 5432 | Justification theorem for the ordered pair definition of Felix Hausdorff in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), 1914, p. 32: 〈𝐴, 𝐵〉H = {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually, any two different fixed sets will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5281). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary in full extent (𝐴 ≠ 𝑇 is not required). This definition is meaningful only for classes 𝐴 and 𝐵 that exist as sets (i.e., are not proper classes): If 𝐴 and 𝐶 were different proper classes (𝐴 ≠ 𝐶), then {{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇} ↔ {{𝑂}, {𝐵, 𝑇}} = {{𝑂}, {𝐷, 𝑇} is true if 𝐵 = 𝐷, but (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) would be false. See df-op 4569 for other ordered pair definitions. (Contributed by AV, 14-Jun-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐴 ≠ 𝑂 & ⊢ 𝐵 ≠ 𝑂 & ⊢ 𝐵 ≠ 𝑇 & ⊢ 𝑂 ∈ V & ⊢ 𝑇 ∈ V & ⊢ 𝑂 ≠ 𝑇 ⇒ ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthhausdorff0 5433 | Justification theorem for the ordered pair definition of Felix Hausdorff in "Grundzüge der Mengenlehre" ("Basics of Set Theory"), 1914, p. 32: 〈𝐴, 𝐵〉H = {{𝐴, 𝑂}, {𝐵, 𝑇}}. Hausdorff used 1 and 2 instead of 𝑂 and 𝑇, but actually, any two different fixed sets will do (e.g., 𝑂 = ∅ and 𝑇 = {∅}, see 0nep0 5281). Furthermore, Hausdorff demanded that 𝑂 and 𝑇 are both different from 𝐴 as well as 𝐵, which is actually not necessary if all involved classes exist as sets (i.e. are not proper classes), in contrast to opthhausdorff 5432. See df-op 4569 for other ordered pair definitions. (Contributed by AV, 12-Jun-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝑂 ∈ V & ⊢ 𝑇 ∈ V & ⊢ 𝑂 ≠ 𝑇 ⇒ ⊢ ({{𝐴, 𝑂}, {𝐵, 𝑇}} = {{𝐶, 𝑂}, {𝐷, 𝑇}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | otsndisj 5434* | The singletons consisting of ordered triples which have distinct third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑉 {〈𝐴, 𝐵, 𝑐〉}) | ||
Theorem | otiunsndisj 5435* | The union of singletons consisting of ordered triples which have distinct first and third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.) |
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑉 ∪ 𝑐 ∈ (𝑊 ∖ {𝑎}){〈𝑎, 𝐵, 𝑐〉}) | ||
Theorem | iunopeqop 5436* | Implication of an ordered pair being equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ ∅ → (∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} = 〈𝐶, 𝐷〉 → ∃𝑧 𝐴 = {𝑧})) | ||
Theorem | brsnop 5437 | Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{〈𝐴, 𝐵〉}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵))) | ||
Theorem | opabidw 5438* | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5439 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 14-Apr-1995.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | opabid 5439 | The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker opabidw 5438 when possible. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) |
⊢ (〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | elopabw 5440* | Membership in a class abstraction of ordered pairs. Weaker version of elopab 5441 with a sethood antecedent, avoiding ax-sep 5224, ax-nul 5231, and ax-pr 5353. Originally a subproof of elopab 5441. (Contributed by SN, 11-Dec-2024.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | ||
Theorem | elopab 5441* | Membership in a class abstraction of ordered pairs. (Contributed by NM, 24-Mar-1998.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | ||
Theorem | rexopabb 5442* | Restricted existential quantification over an ordered-pair class abstraction. (Contributed by AV, 8-Nov-2023.) |
⊢ 𝑂 = {〈𝑥, 𝑦〉 ∣ 𝜑} & ⊢ (𝑜 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒)) | ||
Theorem | vopelopabsb 5443* | The law of concretion in terms of substitutions. Version of opelopabsb 5444 with set variables. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024.) |
⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | opelopabsb 5444* | The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.) |
⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | ||
Theorem | brabsb 5445* | The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | ||
Theorem | opelopabt 5446* | Closed theorem form of opelopab 5456. (Contributed by NM, 19-Feb-2013.) |
⊢ ((∀𝑥∀𝑦(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥∀𝑦(𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | opelopabga 5447* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) | ||
Theorem | brabga 5448* | The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) | ||
Theorem | opelopab2a 5449* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓)) | ||
Theorem | opelopaba 5450* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) | ||
Theorem | braba 5451* | The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝜓) | ||
Theorem | opelopabg 5452* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | brabg 5453* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) | ||
Theorem | opelopabgf 5454* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5452 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | ||
Theorem | opelopab2 5455* | Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜒)) | ||
Theorem | opelopab 5456* | The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) | ||
Theorem | brab 5457* | The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝜒) | ||
Theorem | opelopabaf 5458* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5456 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) | ||
Theorem | opelopabf 5459* | The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5456 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜒 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) | ||
Theorem | ssopab2 5460 | Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓}) | ||
Theorem | ssopab2bw 5461* | Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5463 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 27-Dec-1996.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | eqopab2bw 5462* | Equivalence of ordered pair abstraction equality and biconditional. Version of eqopab2b 5466 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by Mario Carneiro, 4-Jan-2017.) (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | ssopab2b 5463 | Equivalence of ordered pair abstraction subclass and implication. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker ssopab2bw 5461 when possible. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (New usage is discouraged.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | ssopab2i 5464 | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜓} | ||
Theorem | ssopab2dv 5465* | Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ⊆ {〈𝑥, 𝑦〉 ∣ 𝜒}) | ||
Theorem | eqopab2b 5466 | Equivalence of ordered pair abstraction equality and biconditional. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker eqopab2bw 5462 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) | ||
Theorem | opabn0 5467 | Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥∃𝑦𝜑) | ||
Theorem | opab0 5468 | Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.) |
⊢ ({〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | ||
Theorem | csbopab 5469* | Move substitution into a class abstraction. Version of csbopabgALT 5470 without a sethood antecedent but depending on more axioms. (Contributed by NM, 6-Aug-2007.) (Revised by NM, 23-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑} | ||
Theorem | csbopabgALT 5470* | Move substitution into a class abstraction. Version of csbopab 5469 with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{〈𝑦, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ [𝐴 / 𝑥]𝜑}) | ||
Theorem | csbmpt12 5471* | Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) | ||
Theorem | csbmpt2 5472* | Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌 ↦ 𝑍) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝑍)) | ||
Theorem | iunopab 5473* | Move indexed union inside an ordered-pair class abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) Avoid ax-sep 5224, ax-nul 5231, ax-pr 5353. (Revised by SN, 11-Nov-2024.) |
⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} | ||
Theorem | iunopabOLD 5474* | Obsolete version of iunopab 5473 as of 11-Dec-2024. (Contributed by Stefan O'Rear, 20-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∪ 𝑧 ∈ 𝐴 {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 𝜑} | ||
Theorem | elopabr 5475* | Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) | ||
Theorem | elopabran 5476* | Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥𝑅𝑦 ∧ 𝜓)} → 𝐴 ∈ 𝑅) | ||
Theorem | elopabrOLD 5477* | Obsolete version of elopabr 5475 as of 11-Dec-2024. (Contributed by AV, 16-Feb-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅) | ||
Theorem | rbropapd 5478* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))) | ||
Theorem | rbropap 5479* | Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) |
⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒))) | ||
Theorem | 2rbropap 5480* | Properties of a pair in a restricted binary relation 𝑀 expressed as an ordered-pair class abstraction: 𝑀 is the binary relation 𝑊 restricted by the conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.) |
⊢ (𝜑 → 𝑀 = {〈𝑓, 𝑝〉 ∣ (𝑓𝑊𝑝 ∧ 𝜓 ∧ 𝜏)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜏 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒 ∧ 𝜃))) | ||
Theorem | 0nelopab 5481 | The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.) |
⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | ||
Theorem | 0nelopabOLD 5482 | Obsolete version of 0nelopab 5481 as of 3-Oct-2024. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} | ||
Theorem | brabv 5483 | If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
⊢ (𝑋{〈𝑥, 𝑦〉 ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) | ||
Theorem | pwin 5484 | The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) |
⊢ 𝒫 (𝐴 ∩ 𝐵) = (𝒫 𝐴 ∩ 𝒫 𝐵) | ||
Theorem | pwunssOLD 5485 | Obsolete version of pwunss 4554 as of 30-Dec-2023. (Contributed by NM, 23-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵) | ||
Theorem | pwssun 5486 | The power class of the union of two classes is a subset of the union of their power classes, iff one class is a subclass of the other. Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵)) | ||
Theorem | pwundifOLD 5487 | Obsolete proof of pwundif 4560 as of 26-Dec-2023. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴) | ||
Theorem | pwun 5488 | The power class of the union of two classes equals the union of their power classes, iff one class is a subclass of the other. Part of Exercise 7(b) of [Enderton] p. 28. (Contributed by NM, 23-Nov-2003.) |
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) = (𝒫 𝐴 ∪ 𝒫 𝐵)) | ||
Syntax | cid 5489 | Extend the definition of a class to include the identity relation. |
class I | ||
Definition | df-id 5490* | Define the identity relation. Definition 9.15 of [Quine] p. 64. For example, 5 I 5 and ¬ 4 I 5 (ex-id 28807). (Contributed by NM, 13-Aug-1995.) |
⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | ||
Theorem | dfid4 5491 | The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ I = (𝑥 ∈ V ↦ 𝑥) | ||
Theorem | dfid2 5492 |
Alternate definition of the identity relation. Instance of dfid3 5493 not
requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce
axiom usage. (Revised by Gino Giotto, 4-Nov-2024.) (Proof shortened by
BJ, 5-Nov-2024.)
Use df-id 5490 instead to make the semantics of the constructor df-opab 5138 clearer. (New usage is discouraged.) |
⊢ I = {〈𝑥, 𝑥〉 ∣ 𝑥 = 𝑥} | ||
Theorem | dfid3 5493 |
A stronger version of df-id 5490 that does not require 𝑥 and 𝑦 to
be disjoint. This is not the definition since, in order to pass our
definition soundness test, a definition has to have disjoint dummy
variables, see conventions 28773. The proof can be instructive in
showing
how disjoint variable conditions may be eliminated, a task that is not
necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by
Mario Carneiro, 18-Nov-2016.)
Use df-id 5490 instead to make the semantics of the constructor df-opab 5138 clearer (in usages, 𝑥, 𝑦 will typically be dummy variables, so can be assumed disjoint). (New usage is discouraged.) |
⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | ||
Theorem | dfid2OLD 5494 | Obsolete version of dfid2 5492 as of 4-Nov-2024. (Contributed by NM, 15-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ I = {〈𝑥, 𝑥〉 ∣ 𝑥 = 𝑥} | ||
Syntax | cep 5495 | Extend class notation to include the membership relation. |
class E | ||
Definition | df-eprel 5496* | Define the membership relation (also called "epsilon relation" since it is sometimes denoted by the lowercase Greek letter "epsilon"). Similar to Definition 6.22 of [TakeutiZaring] p. 30. The membership relation and the membership predicate agree, that is, (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵), when 𝐵 is a set (see epelg 5497). Thus, ⊢ 5 E {1, 5} (ex-eprel 28806). (Contributed by NM, 13-Aug-1995.) |
⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | ||
Theorem | epelg 5497 | The membership relation and the membership predicate agree when the "containing" class is a set. General version of epel 5499 and closed form of epeli 5498. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof shortened by BJ, 14-Jul-2023.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | ||
Theorem | epeli 5498 | The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5497. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | ||
Theorem | epel 5499 | The membership relation and the membership predicate agree when the "containing" class is a setvar. (Contributed by NM, 13-Aug-1995.) Replace the first setvar variable with a class variable. (Revised by BJ, 13-Sep-2022.) |
⊢ (𝐴 E 𝑥 ↔ 𝐴 ∈ 𝑥) | ||
Theorem | 0sn0ep 5500 | An example for the membership relation. (Contributed by AV, 19-Jun-2022.) |
⊢ ∅ E {∅} |
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