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Theorem List for Metamath Proof Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremraldifsni 4701 Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
(∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵))

Theoremraldifsnb 4702* Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)

Theoremeldifvsn 4703 A set is an element of the universal class excluding a singleton iff it is not the singleton element. (Contributed by AV, 7-Apr-2019.)
(𝐴𝑉 → (𝐴 ∈ (V ∖ {𝐵}) ↔ 𝐴𝐵))

Theoremdifsn 4704 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 4705 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 4706 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Theoremdifprsn2 4707 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})

Theoremdiftpsn3 4708 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Proof shortened by JJ, 23-Jul-2021.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})

Theoremdifpr 4709 Removing two elements as pair of elements corresponds to removing each of the two elements as singletons. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
(𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})

Theoremtpprceq3 4710 An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
(¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Theoremtppreqb 4711 An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)
(¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Theoremdifsnb 4712 (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4704. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Theoremdifsnpss 4713 (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if and only if 𝐴 is a member of 𝐵. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Theoremsnssi 4714 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

Theoremsnssd 4715 The singleton of an element of a class is a subset of the class (deduction form). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)       (𝜑 → {𝐴} ⊆ 𝐵)

Theoremdifsnid 4716 If we remove a single element from a class then put it back in, we end up with the original class. (Contributed by NM, 2-Oct-2006.)
(𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴)

Theoremeldifeldifsn 4717 An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022.)
((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋}))

Theorempw0 4718 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 ∅ = {∅}

Theorempwpw0 4719 Compute the power set of the power set of the empty set. (See pw0 4718 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4805, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.)
𝒫 {∅} = {∅, {∅}}

Theoremsnsspr1 4720 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
{𝐴} ⊆ {𝐴, 𝐵}

Theoremsnsspr2 4721 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
{𝐵} ⊆ {𝐴, 𝐵}

Theoremsnsstp1 4722 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐴} ⊆ {𝐴, 𝐵, 𝐶}

Theoremsnsstp2 4723 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐵} ⊆ {𝐴, 𝐵, 𝐶}

Theoremsnsstp3 4724 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐶} ⊆ {𝐴, 𝐵, 𝐶}

Theoremprssg 4725 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))

Theoremprss 4726 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.) (Proof shortened by JJ, 23-Jul-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴𝐶𝐵𝐶) ↔ {𝐴, 𝐵} ⊆ 𝐶)

Theoremprssi 4727 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)

Theoremprssd 4728 Deduction version of prssi 4727: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)

Theoremprsspwg 4729 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)))

Theoremssprss 4730 A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐴 = 𝐷) ∧ (𝐵 = 𝐶𝐵 = 𝐷))))

Theoremssprsseq 4731 A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐶, 𝐷}))

Theoremsssn 4732 The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)
(𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))

Theoremssunsn2 4733 The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4808. (Contributed by Mario Carneiro, 2-Jul-2016.)
((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))

Theoremssunsn 4734 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))

Theoremeqsn 4735* Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.) (Proof shortened by JJ, 23-Jul-2021.)
(𝐴 ≠ ∅ → (𝐴 = {𝐵} ↔ ∀𝑥𝐴 𝑥 = 𝐵))

Theoremissn 4736* A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)
(∃𝑥𝐴𝑦𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})

Theoremn0snor2el 4737* A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)
(𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))

Theoremssunpr 4738 Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))

Theoremsspr 4739 The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)
(𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))

Theoremsstp 4740 The subsets of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
(𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))

Theoremtpss 4741 An unordered triple 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 4742 An unordered 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)

Theoremsneqrg 4743 Closed form of sneqr 4744. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.)
(𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Theoremsneqr 4744 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 4745 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
𝐴 ∈ V       ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Theoremmosneq 4746* There exists at most one set whose singleton is equal to a given class. See also moeq 3673. (Contributed by BJ, 24-Sep-2022.)
∃*𝑥{𝑥} = 𝐴

Theoremsneqbg 4747 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
(𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Theoremsnsspw 4748 The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.)
{𝐴} ⊆ 𝒫 𝐴

Theoremprsspw 4749 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.) (Proof shortened by OpenAI, 25-Mar-2020.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴𝐶𝐵𝐶))

Theorempreq1b 4750 Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same second element iff the first elements are equal. (Contributed by AV, 18-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ({𝐴, 𝐶} = {𝐵, 𝐶} ↔ 𝐴 = 𝐵))

Theorempreq2b 4751 Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ({𝐶, 𝐴} = {𝐶, 𝐵} ↔ 𝐴 = 𝐵))

Theorempreqr1 4752 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 4753 Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵)

Theorempreq12b 4754 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)))

Theoremopthpr 4755 An unordered pair has the ordered pair property (compare opth 5345) under certain conditions. (Contributed by NM, 27-Mar-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (𝐴𝐷 → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempreqr1g 4756 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4752. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))

Theorempreq12bg 4757 Closed form of preq12b 4754. (Contributed by Scott Fenton, 28-Mar-2014.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))

Theoremprneimg 4758 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}))

Theoremprnebg 4759 A (proper) pair is not equal to another (maybe improper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018.)
(((𝐴𝑈𝐵𝑉) ∧ (𝐶𝑋𝐷𝑌) ∧ 𝐴𝐵) → (((𝐴𝐶𝐴𝐷) ∨ (𝐵𝐶𝐵𝐷)) ↔ {𝐴, 𝐵} ≠ {𝐶, 𝐷}))

Theorempr1eqbg 4760 A (proper) pair is equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴 = 𝐶 ↔ {𝐴, 𝐵} = {𝐵, 𝐶}))

Theorempr1nebg 4761 A (proper) pair is not equal to another (maybe improper) pair containing one element of the first pair if and only if the other element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(((𝐴𝑈𝐵𝑉𝐶𝑋) ∧ 𝐴𝐵) → (𝐴𝐶 ↔ {𝐴, 𝐵} ≠ {𝐵, 𝐶}))

Theorempreqsnd 4762 Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Revised by AV, 13-Jun-2022.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)       (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶𝐵 = 𝐶)))

Theoremprnesn 4763 A proper unordered pair is not a (proper or improper) singleton. (Contributed by AV, 13-Jun-2022.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝐴, 𝐵} ≠ {𝐶})

Theoremprneprprc 4764 A proper unordered pair is not an improper unordered pair. (Contributed by AV, 13-Jun-2022.)
(((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})

Theorempreqsn 4765 Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 12-Jun-2022.)
𝐴 ∈ V    &   𝐵 ∈ V       ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Theorempreq12nebg 4766 Equality relationship for two proper unordered pairs. (Contributed by AV, 12-Jun-2022.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))

Theoremprel12g 4767 Equality of two unordered pairs. (Contributed by NM, 17-Oct-1996.) (Revised by AV, 9-Dec-2018.) (Revised by AV, 12-Jun-2022.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷})))

Theoremopthprneg 4768 An unordered pair has the ordered pair property (compare opth 5345) under certain conditions. Variant of opthpr 4755 in closed form. (Contributed by AV, 13-Jun-2022.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremelpreqprlem 4769* Lemma for elpreqpr 4770. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)
(𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})

Theoremelpreqpr 4770* Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)
(𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})

Theoremelpreqprb 4771* A set is an element of an unordered pair iff there is another (maybe the same) set which is an element of the unordered pair. (Proposed by BJ, 8-Dec-2020.) (Contributed by AV, 9-Dec-2020.)
(𝐴𝑉 → (𝐴 ∈ {𝐵, 𝐶} ↔ ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))

Theoremelpr2elpr 4772* For an element 𝐴 of an unordered pair which is a subset of a given set 𝑉, there is another (maybe the same) element 𝑏 of the given set 𝑉 being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
((𝑋𝑉𝑌𝑉𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Theoremdfopif 4773 Rewrite df-op 4546 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)

Theoremdfopg 4774 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})

Theoremdfop 4775 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.) (Avoid depending on this detail.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}

Theoremopeq1 4776 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-10 2145, ax-11 2161, ax-12 2178. (Revised by Gino Giotto, 26-May-2024.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq1OLD 4777 Obsolete version of opeq1 4776 as of 25-May-2024. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq2 4778 Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-10 2145, ax-11 2161, ax-12 2178. (Revised by Gino Giotto, 26-May-2024.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq2OLD 4779 Obsolete version of opeq2 4778 as of 25-May-2024. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq12 4780 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)

Theoremopeq1i 4781 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶

Theoremopeq2i 4782 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵

Theoremopeq12i 4783 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 4784 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Theoremopeq2d 4785 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)

Theoremopeq12d 4786 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 4787 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Theoremoteq2 4788 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Theoremoteq3 4789 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Theoremoteq1d 4790 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Theoremoteq2d 4791 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Theoremoteq3d 4792 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Theoremoteq123d 4793 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 4794 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵

Theoremnfopd 4795 Deduction version of bound-variable hypothesis builder nfop 4794. This shows how the deduction version of a not-free theorem such as nfop 4794 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)

Theoremcsbopg 4796 Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.)
(𝐴𝑉𝐴 / 𝑥𝐶, 𝐷⟩ = ⟨𝐴 / 𝑥𝐶, 𝐴 / 𝑥𝐷⟩)

Theoremopidg 4797 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Closed form of opid 4798. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.)
(𝐴𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})

Theoremopid 4798 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. Inference form of opidg 4797. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (Avoid depending on this detail.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}

Theoremralunsn 4799* 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 4800* 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))))

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