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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremelALT 4901* Alternate proof of el 4838, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑦 𝑥𝑦

TheoremdtruALT2 4902* Alternate proof of dtru 4848 using ax-pr 4897 instead of ax-pow 4834. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦

Theoremsnelpwi 4903 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Theoremsnelpw 4904 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
𝐴 ∈ V       (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Theoremprelpw 4905 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.)
((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))

Theoremprelpwi 4906 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.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)

Theoremrext 4907* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
(∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)

Theoremsspwb 4908 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)

Theoremunipw 4909 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

Theoremuniv 4910 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
V = V

Theorempwel 4911 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
(𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)

Theorempwtr 4912 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)

Theoremssextss 4913* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
(𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremssext 4914* 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(𝑥𝐴𝑥𝐵))

Theoremnssss 4915* Negation of subclass relationship. Compare nss 3655. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐴𝐵 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵))

Theorempweqb 4916 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)

Theoremintid 4917* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
𝐴 ∈ V        {𝑥𝐴𝑥} = {𝐴}

Theoremmoabex 4918 "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
(∃*𝑥𝜑 → {𝑥𝜑} ∈ V)

Theoremrmorabex 4919 Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
(∃*𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ∈ V)

Theoremeuabex 4920 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
(∃!𝑥𝜑 → {𝑥𝜑} ∈ V)

Theoremnnullss 4921* A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
(𝐴 ≠ ∅ → ∃𝑥(𝑥𝐴𝑥 ≠ ∅))

Theoremexss 4922* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
(∃𝑥𝐴 𝜑 → ∃𝑦(𝑦𝐴 ∧ ∃𝑥𝑦 𝜑))

Theoremopex 4923 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

Theoremotex 4924 An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
𝐴, 𝐵, 𝐶⟩ ∈ V

Theoremelopg 4925 Characterization of the elements of an ordered pair. Closed form of elop 4926. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))

Theoremelop 4926 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}))

TheoremelopOLD 4927 Obsolete version of elop 4926, with one extraneous hypothesis. Obsolete as of 25-Dec-2020 . (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 ∈ V    &   𝐶 ∈ V    &   𝐴 ∈ V       (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))

Theoremopi1 4928 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

Theoremopi2 4929 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

Theoremopeluu 4930 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 𝐶𝐵 𝐶))

Theoremop1stb 4931 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 5607 to extract the second member, op1sta 5605 for an alternate version, and op1st 7161 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝐴, 𝐵⟩ = 𝐴

2.3.3  Ordered pair theorem

Theoremopnz 4932 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))

Theoremopnzi 4933 An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ ≠ ∅

Theoremopth1 4934 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)

Theoremopth 4935 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 = 𝐶𝐵 = 𝐷))

Theoremopthg 4936 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.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremopth1g 4937 Equality of the first members of equal ordered pairs. Closed form of opth1 4934. (Contributed by AV, 14-Oct-2018.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶))

Theoremopthg2 4938 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐶𝑉𝐷𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremopth2 4939 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
𝐶 ∈ V    &   𝐷 ∈ V       (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremopthneg 4940 Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝐶𝐵𝐷)))

Theoremopthne 4941 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⟩ ↔ (𝐴𝐶𝐵𝐷))

Theoremotth2 4942 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 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))

Theoremotth 4943 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 = 𝐶𝐵 = 𝐷𝑅 = 𝑆))

Theoremotthg 4944 Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
((𝐴𝑈𝐵𝑉𝐶𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))

Theoremeqvinop 4945* 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⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))

Theoremcopsexg 4946* 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 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Theoremcopsex2t 4947* Closed theorem form of copsex2g 4948. (Contributed by NM, 17-Feb-2013.)
((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓)) ∧ (𝐴𝑉𝐵𝑊)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

Theoremcopsex2g 4948* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

Theoremcopsex4g 4949* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) → (𝜑𝜓))       (((𝐴𝑅𝐵𝑆) ∧ (𝐶𝑅𝐷𝑆)) → (∃𝑥𝑦𝑧𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ 𝜓))

Theorem0nelop 4950 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
¬ ∅ ∈ ⟨𝐴, 𝐵

Theoremopwo0id 4951 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⟩ ∖ {∅})

Theoremopeqex 4952 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)))

Theoremoteqex2 4953 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))

Theoremoteqex 4954 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)))

Theoremopcom 4955 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)

Theoremmoop2 4956* "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

Theoremopeqsn 4957 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))

Theoremopeqpr 4958 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})))

Theoremsnopeqop 4959 Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))

Theorempropeqop 4960 Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V       ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷𝐹 = {𝐴, 𝐵}))))

Theorempropssopi 4961 If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V       ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)

Theoremmosubopt 4962* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
(∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

Theoremmosubop 4963* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)

Theoremeuop2 4964* Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
𝐴 ∈ V       (∃!𝑥𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)

Theoremeuotd 4965* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝜓 ↔ (𝑎 = 𝐴𝑏 = 𝐵𝑐 = 𝐶)))       (𝜑 → ∃!𝑥𝑎𝑏𝑐(𝑥 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ 𝜓))

Theoremopthwiener 4966 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 4175 for other ordered pair definitions. (Contributed by NM, 28-Sep-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       ({{{𝐴}, ∅}, {{𝐵}}} = {{{𝐶}, ∅}, {{𝐷}}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremuniop 4967 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}

Theoremuniopel 4968 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⟩ ∈ 𝐶𝐴, 𝐵⟩ ∈ 𝐶)

Theoremotsndisj 4969* The singletons consisting of ordered triples which have distinct third components are disjoint. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑉 {⟨𝐴, 𝐵, 𝑐⟩})

Theoremotiunsndisj 4970* 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.)
(𝐵𝑋Disj 𝑎𝑉 𝑐 ∈ (𝑊 ∖ {𝑎}){⟨𝑎, 𝐵, 𝑐⟩})

Theoremiunopeqop 4971* Implication of an ordered pair being equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.)
𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴 ≠ ∅ → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))

2.3.4  Ordered-pair class abstractions (cont.)

Theoremopabid 4972 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)

Theoremelopab 4973* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))

TheoremopelopabsbALT 4974* The law of concretion in terms of substitutions. Less general than opelopabsb 4975, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)

Theoremopelopabsb 4975* 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)

Theorembrabsb 4976* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)

Theoremopelopabt 4977* Closed theorem form of opelopab 4987. (Contributed by NM, 19-Feb-2013.)
((∀𝑥𝑦(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝑦(𝑦 = 𝐵 → (𝜓𝜒)) ∧ (𝐴𝑉𝐵𝑊)) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Theoremopelopabga 4978* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓))

Theorembrabga 4979* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝜓))

Theoremopelopab2a 4980* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))

Theoremopelopaba 4981* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

Theorembraba 4982* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵𝜓)

Theoremopelopabg 4983* 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 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Theorembrabg 4984* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))

Theoremopelopabgf 4985* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4983 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
𝑥𝜓    &   𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Theoremopelopab2 4986* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜒))

Theoremopelopab 4987* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Theorembrab 4988* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵𝜒)

Theoremopelopabaf 4989* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4987 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.)
𝑥𝜓    &   𝑦𝜓    &   𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

Theoremopelopabf 4990* The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4987 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
𝑥𝜓    &   𝑦𝜒    &   𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))       (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Theoremssopab2 4991 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 19-May-2013.)
(∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})

Theoremssopab2b 4992 Equivalence of ordered pair abstraction subclass and implication. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremssopab2i 4993 Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Theoremssopab2dv 4994* Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Theoremeqopab2b 4995 Equivalence of ordered pair abstraction equality and biconditional. (Contributed by Mario Carneiro, 4-Jan-2017.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremopabn0 4996 Nonempty ordered pair class abstraction. (Contributed by NM, 10-Oct-2007.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝑦𝜑)

Theoremopab0 4997 Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥𝑦 ¬ 𝜑)

Theoremcsbopab 4998* Move substitution into a class abstraction. Version of csbopabgALT 4999 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}

TheoremcsbopabgALT 4999* Move substitution into a class abstraction. Version of csbopab 4998 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.)
(𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})

Theoremcsbmpt12 5000* Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.)
(𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))

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