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Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtpidm23 3501 Unordered triple {𝐴, 𝐵, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.)
{𝐴, 𝐵, 𝐵} = {𝐴, 𝐵}

Theoremtpidm 3502 Unordered triple {𝐴, 𝐴, 𝐴} is just an overlong way to write {𝐴}. (Contributed by David A. Wheeler, 10-May-2015.)
{𝐴, 𝐴, 𝐴} = {𝐴}

Theoremtppreq3 3503 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
(𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Theoremprid1g 3504 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
(𝐴𝑉𝐴 ∈ {𝐴, 𝐵})

Theoremprid2g 3505 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)
(𝐵𝑉𝐵 ∈ {𝐴, 𝐵})

Theoremprid1 3506 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       𝐴 ∈ {𝐴, 𝐵}

Theoremprid2 3507 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
𝐵 ∈ V       𝐵 ∈ {𝐴, 𝐵}

Theoremprprc1 3508 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V → {𝐴, 𝐵} = {𝐵})

Theoremprprc2 3509 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
𝐵 ∈ V → {𝐴, 𝐵} = {𝐴})

Theoremprprc 3510 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)
((¬ 𝐴 ∈ V ∧ ¬ 𝐵 ∈ V) → {𝐴, 𝐵} = ∅)

Theoremtpid1 3511 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V       𝐴 ∈ {𝐴, 𝐵, 𝐶}

Theoremtpid2 3512 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐵 ∈ V       𝐵 ∈ {𝐴, 𝐵, 𝐶}

Theoremtpid3g 3513 Closed theorem form of tpid3 3514. (Contributed by Alan Sare, 24-Oct-2011.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Theoremtpid3 3514 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐶 ∈ V       𝐶 ∈ {𝐴, 𝐵, 𝐶}

Theoremsnnzg 3515 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
(𝐴𝑉 → {𝐴} ≠ ∅)

Theoremsnmg 3516* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
(𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})

Theoremsnnz 3517 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴} ≠ ∅

Theoremsnm 3518* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
𝐴 ∈ V       𝑥 𝑥 ∈ {𝐴}

Theoremprmg 3519* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
(𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})

Theoremprnz 3520 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
𝐴 ∈ V       {𝐴, 𝐵} ≠ ∅

Theoremprm 3521* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
𝐴 ∈ V       𝑥 𝑥 ∈ {𝐴, 𝐵}

Theoremprnzg 3522 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
(𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Theoremtpnz 3523 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴, 𝐵, 𝐶} ≠ ∅

Theoremsnss 3524 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V       (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵)

Theoremeldifsn 3525 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵𝐴𝐶))

Theoremeldifsni 3526 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)

Theoremneldifsn 3527 𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
¬ 𝐴 ∈ (𝐵 ∖ {𝐴})

Theoremneldifsnd 3528 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))

Theoremrexdifsn 3529 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))

Theoremsnssg 3530 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)
(𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))

Theoremdifsn 3531 An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Theoremdifprsnss 3532 Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
({𝐴, 𝐵} ∖ {𝐴}) ⊆ {𝐵}

Theoremdifprsn1 3533 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Theoremdifprsn2 3534 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Theoremdiftpsn3 3535 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})

Theoremdifsnb 3536 (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3531. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Theoremsnssi 3537 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

Theoremsnssd 3538 The singleton of an element of a class is a subset of the class (deduction rule). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → {𝐴} ⊆ 𝐵)

Theoremdifsnss 3539 If we remove a single element from a class then put it back in, we end up with a subset of the original class. If equality is decidable, we can replace subset with equality as seen in nndifsnid 6146. (Contributed by Jim Kingdon, 10-Aug-2018.)
(𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)

Theorempw0 3540 Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝒫 ∅ = {∅}

Theoremsnsspr1 3541 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
{𝐴} ⊆ {𝐴, 𝐵}

Theoremsnsspr2 3542 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
{𝐵} ⊆ {𝐴, 𝐵}

Theoremsnsstp1 3543 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Theoremsnsstp2 3544 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Theoremsnsstp3 3545 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Theoremprsstp12 3546 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
{𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Theoremprsstp13 3547 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
{𝐴, 𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Theoremprsstp23 3548 A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)
{𝐵, 𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Theoremprss 3549 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Theoremprssg 3550 A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
((𝐴𝑉𝐵𝑊) → ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶))

Theoremprssi 3551 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)

Theoremprsspwg 3552 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Theoremsssnr 3553 Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})

Theoremsssnm 3554* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
(∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))

Theoremeqsnm 3555* Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
(∃𝑥 𝑥𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))

Theoremssprr 3556 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
(((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})

Theoremsstpr 3557 The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
((((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})

Theoremtpss 3558 A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((𝐴𝐷𝐵𝐷𝐶𝐷) ↔ {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Theoremtpssi 3559 A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
((𝐴𝐷𝐵𝐷𝐶𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Theoremsneqr 3560 If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)
𝐴 ∈ V       ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Theoremsnsssn 3561 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
𝐴 ∈ V       ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Theoremsneqrg 3562 Closed form of sneqr 3560. (Contributed by Scott Fenton, 1-Apr-2011.)
(𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Theoremsneqbg 3563 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
(𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Theoremsnsspw 3564 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
{𝐴} ⊆ 𝒫 𝐴

Theoremprsspw 3565 An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶))

Theorempreqr1g 3566 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3568. (Contributed by Jim Kingdon, 21-Sep-2018.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Theorempreqr2g 3567 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3569. (Contributed by Jim Kingdon, 21-Sep-2018.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵))

Theorempreqr1 3568 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵)

Theorempreqr2 3569 Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Theorempreq12b 3570 Equality relationship for two unordered pairs. (Contributed by NM, 17-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))

Theoremprel12 3571 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       𝐴 = 𝐵 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Theoremopthpr 3572 A way to represent ordered pairs using unordered pairs with distinct members. (Contributed by NM, 27-Mar-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempreq12bg 3573 Closed form of preq12b 3570. (Contributed by Scott Fenton, 28-Mar-2014.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))

Theoremprneimg 3574 Two pairs are not equal if at least one element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
(((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌)) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Theorempreqsn 3575 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Theoremdfopg 3576 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Theoremdfop 3577 Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Theoremopeq1 3578 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq2 3579 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq12 3580 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Theoremopeq1i 3581 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶

Theoremopeq2i 3582 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵

Theoremopeq12i 3583 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴, 𝐶⟩ = ⟨𝐵, 𝐷

Theoremopeq1d 3584 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq2d 3585 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq12d 3586 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)

Theoremoteq1 3587 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Theoremoteq2 3588 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Theoremoteq3 3589 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Theoremoteq1d 3590 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Theoremoteq2d 3591 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Theoremoteq3d 3592 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Theoremoteq123d 3593 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐸 = 𝐹)       (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Theoremnfop 3594 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵

Theoremnfopd 3595 Deduction version of bound-variable hypothesis builder nfop 3594. This shows how the deduction version of a not-free theorem such as nfop 3594 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)

Theoremopid 3596 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}

Theoremralunsn 3597* Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)
(𝑥 = 𝐵 → (𝜑𝜓))       (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥𝐴 𝜑𝜓)))

Theorem2ralunsn 3598* Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)
(𝑥 = 𝐵 → (𝜑𝜒))    &   (𝑦 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜓𝜃))       (𝐵𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥𝐴𝑦𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ (∀𝑦𝐴 𝜒𝜃))))

Theoremopprc 3599 Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
(¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)

Theoremopprc1 3600 Expansion of an ordered pair when the first member is a proper class. See also opprc 3599. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

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