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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rexxfrd 5301* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfr2d 5302* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rexxfr2d 5303* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfrd2 5304* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5300. (Contributed by Alexander van der Vekens, 25-Apr-2018.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | rexxfrd2 5305* | Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 5301. (Contributed by Alexander van der Vekens, 25-Apr-2018.) |
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒)) | ||
Theorem | ralxfr 5306* | Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) | ||
Theorem | ralxfrALT 5307* | Alternate proof of ralxfr 5306 which does not use ralxfrd 5300. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓) | ||
Theorem | rexxfr 5308* | Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵) & ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓) | ||
Theorem | rabxfrd 5309* | Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (Contributed by NM, 16-Jan-2012.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒})) | ||
Theorem | rabxfr 5310* | Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑦𝐶 & ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶) ⇒ ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓})) | ||
Theorem | reuhypd 5311* | A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7137. (Contributed by NM, 16-Jan-2012.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | ||
Theorem | reuhyp 5312* | A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3742. (Contributed by NM, 15-Nov-2004.) |
⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) & ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) ⇒ ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | ||
Theorem | zfpair 5313 |
The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of
[TakeutiZaring] p. 15. In some
textbooks this is stated as a separate
axiom; here we show it is redundant since it can be derived from the
other axioms.
This theorem should not be referenced by any proof other than axprALT 5314. Instead, use zfpair2 5322 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | axprALT 5314* | Alternate proof of axpr 5320. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Theorem | axprlem1 5315* | Lemma for axpr 5320. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥) | ||
Theorem | axprlem2 5316* | Lemma for axpr 5320. There exists a set to which all sets whose only members are empty sets belong. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥) | ||
Theorem | axprlem3 5317* | Lemma for axpr 5320. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.) |
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) | ||
Theorem | axprlem4 5318* | Lemma for axpr 5320. The first element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) | ||
Theorem | axprlem5 5319* | Lemma for axpr 5320. The second element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) |
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))) | ||
Theorem | axpr 5320* |
Unabbreviated version of the Axiom of Pairing of ZF set theory, derived
as a theorem from the other axioms.
This theorem should not be referenced by any proof. Instead, use ax-pr 5321 below so that the uses of the Axiom of Pairing can be more easily identified. For a shorter proof using ax-ext 2793, see axprALT 5314. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2793. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) Use ax-pr 5321 instead. (New usage is discouraged.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Axiom | ax-pr 5321* | The Axiom of Pairing of ZF set theory. It was derived as theorem axpr 5320 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 14-Nov-2006.) |
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) | ||
Theorem | zfpair2 5322 | Derive the abbreviated version of the Axiom of Pairing from ax-pr 5321. See zfpair 5313 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.) |
⊢ {𝑥, 𝑦} ∈ V | ||
Theorem | snex 5323 | A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 5275. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.) |
⊢ {𝐴} ∈ V | ||
Theorem | prex 5324 | The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 4697), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.) |
⊢ {𝐴, 𝐵} ∈ V | ||
Theorem | sels 5325* | If a class is a set, then it is a member of a set. (Contributed by BJ, 3-Apr-2019.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) | ||
Theorem | elALT 5326* | Alternate proof of el 5262, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑦 𝑥 ∈ 𝑦 | ||
Theorem | dtruALT2 5327* | Alternate proof of dtru 5263 using ax-pr 5321 instead of ax-pow 5258. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | snelpwi 5328 | A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.) |
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵) | ||
Theorem | snelpw 5329 | A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵) | ||
Theorem | prelpw 5330 | A pair of two sets belongs to the power class of a class containing those two sets and vice versa. (Contributed by AV, 8-Jan-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶)) | ||
Theorem | prelpwi 5331 | A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶) | ||
Theorem | rext 5332* | A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.) |
⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦) | ||
Theorem | sspwb 5333 | The powerclass construction preserves and reflects inclusion. Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵) | ||
Theorem | unipw 5334 | A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.) |
⊢ ∪ 𝒫 𝐴 = 𝐴 | ||
Theorem | univ 5335 | The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
⊢ ∪ V = V | ||
Theorem | pwel 5336 | Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) |
⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | ||
Theorem | pwtr 5337 | A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.) |
⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴) | ||
Theorem | ssextss 5338* | An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.) |
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵)) | ||
Theorem | ssext 5339* | An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.) |
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵)) | ||
Theorem | nssss 5340* | Negation of subclass relationship. Compare nss 4028. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵)) | ||
Theorem | pweqb 5341 | Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵) | ||
Theorem | intid 5342* | The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴} | ||
Theorem | moabex 5343 | "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.) |
⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | rmorabex 5344 | Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.) |
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | ||
Theorem | euabex 5345 | The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) | ||
Theorem | nnullss 5346* | A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.) |
⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)) | ||
Theorem | exss 5347* | Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.) |
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑)) | ||
Theorem | opex 5348 | An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 〈𝐴, 𝐵〉 ∈ V | ||
Theorem | otex 5349 | An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.) |
⊢ 〈𝐴, 𝐵, 𝐶〉 ∈ V | ||
Theorem | elopg 5350 | Characterization of the elements of an ordered pair. Closed form of elop 5351. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 〈𝐴, 𝐵〉 ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))) | ||
Theorem | elop 5351 | Characterization of the elements of an ordered pair. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) Remove an extraneous hypothesis. (Revised by BJ, 25-Dec-2020.) (Avoid depending on this detail.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 ∈ 〈𝐵, 𝐶〉 ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶})) | ||
Theorem | opi1 5352 | One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {𝐴} ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opi2 5353 | One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opeluu 5354 | Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶)) | ||
Theorem | op1stb 5355 | Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6078 to extract the second member, op1sta 6076 for an alternate version, and op1st 7688 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 | ||
Theorem | brv 5356 | Two classes are always in relation by V. This is simply equivalent to 〈𝐴, 𝐵〉 ∈ V, and does not imply that V is a relation: see nrelv 5667. (Contributed by Scott Fenton, 11-Apr-2012.) |
⊢ 𝐴V𝐵 | ||
Theorem | opnz 5357 | An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (〈𝐴, 𝐵〉 ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | opnzi 5358 | An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ 〈𝐴, 𝐵〉 ≠ ∅ | ||
Theorem | opth1 5359 | Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶) | ||
Theorem | opth 5360 | The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthg 5361 | Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | opth1g 5362 | Equality of the first members of equal ordered pairs. Closed form of opth1 5359. (Contributed by AV, 14-Oct-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → 𝐴 = 𝐶)) | ||
Theorem | opthg2 5363 | Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | opth2 5364 | Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthneg 5365 | Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))) | ||
Theorem | opthne 5366 | Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 ≠ 〈𝐶, 𝐷〉 ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)) | ||
Theorem | otth2 5367 | Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑅 ∈ V ⇒ ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) | ||
Theorem | otth 5368 | Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝑅 ∈ V ⇒ ⊢ (〈𝐴, 𝐵, 𝑅〉 = 〈𝐶, 𝐷, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) | ||
Theorem | otthg 5369 | 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 5370* | 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 5371* | 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 5372* | 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 5373* | Version of copsexg 5374 with a disjoint variable condition, which does not require ax-13 2383. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | ||
Theorem | copsexg 5374* | Substitution of class 𝐴 for ordered pair 〈𝑥, 𝑦〉. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 25-Aug-2019.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑))) | ||
Theorem | copsex2t 5375* | Closed theorem form of copsex2g 5376. (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 5376* | Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ 𝜓)) | ||
Theorem | copsex4g 5377* | 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 5378 | A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 | ||
Theorem | opwo0id 5379 | 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 5380 | 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 5381 | 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 5382 | 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 5383 | 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 5384* | "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 5385 | 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 5386 | 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 5387 | 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 5388 | 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 5389 | 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 5390 | 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 5391 | 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 5392* | "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 5393* | "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 5394* | 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 5395* | Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝜓 ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ∧ 𝑐 = 𝐶))) ⇒ ⊢ (𝜑 → ∃!𝑥∃𝑎∃𝑏∃𝑐(𝑥 = 〈𝑎, 𝑏, 𝑐〉 ∧ 𝜓)) | ||
Theorem | opthwiener 5396 | Justification theorem for the ordered pair definition in Norbert Wiener, "A simplification of the logic of relations", Proc. of the Cambridge Philos. Soc., 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 4566 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 5397 | 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 5398 | 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 5399 | 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 5250). 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 4566 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 5400 | 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 5250). 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 5399. See df-op 4566 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)) |
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