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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtpid1g 3601 Closed theorem form of tpid1 3600. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴𝐵𝐴 ∈ {𝐴, 𝐶, 𝐷})
 
Theoremtpid2 3602 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}
 
Theoremtpid2g 3603 Closed theorem form of tpid2 3602. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐴, 𝐷})
 
Theoremtpid3g 3604 Closed theorem form of tpid3 3605. (Contributed by Alan Sare, 24-Oct-2011.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremtpid3 3605 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 3606 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)
(𝐴𝑉 → {𝐴} ≠ ∅)
 
Theoremsnmg 3607* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
(𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
 
Theoremsnnz 3608 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴} ≠ ∅
 
Theoremsnm 3609* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)
𝐴 ∈ V       𝑥 𝑥 ∈ {𝐴}
 
Theoremprmg 3610* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
(𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴, 𝐵})
 
Theoremprnz 3611 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
𝐴 ∈ V       {𝐴, 𝐵} ≠ ∅
 
Theoremprm 3612* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)
𝐴 ∈ V       𝑥 𝑥 ∈ {𝐴, 𝐵}
 
Theoremprnzg 3613 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
(𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
 
Theoremtpnz 3614 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)
𝐴 ∈ V       {𝐴, 𝐵, 𝐶} ≠ ∅
 
Theoremsnss 3615 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 3616 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵𝐴𝐶))
 
Theoremssdifsn 3617 Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)
(𝐴 ⊆ (𝐵 ∖ {𝐶}) ↔ (𝐴𝐵 ∧ ¬ 𝐶𝐴))
 
Theoremeldifsni 3618 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
(𝐴 ∈ (𝐵 ∖ {𝐶}) → 𝐴𝐶)
 
Theoremneldifsn 3619 𝐴 is not in (𝐵 ∖ {𝐴}). (Contributed by David Moews, 1-May-2017.)
¬ 𝐴 ∈ (𝐵 ∖ {𝐴})
 
Theoremneldifsnd 3620 𝐴 is not in (𝐵 ∖ {𝐴}). Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
 
Theoremrexdifsn 3621 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ↔ ∃𝑥𝐴 (𝑥𝐵𝜑))
 
Theoremsnssg 3622 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 3623 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 3624 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 3625 Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
 
Theoremdifprsn2 3626 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
 
Theoremdiftpsn3 3627 Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
 
Theoremdifpr 3628 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})
 
Theoremdifsnb 3629 (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3623. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
 
Theoremsnssi 3630 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)
 
Theoremsnssd 3631 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)
 
Theoremdifsnss 3632 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 6357. (Contributed by Jim Kingdon, 10-Aug-2018.)
(𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
 
Theorempw0 3633 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 3634 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
{𝐴} ⊆ {𝐴, 𝐵}
 
Theoremsnsspr2 3635 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
{𝐵} ⊆ {𝐴, 𝐵}
 
Theoremsnsstp1 3636 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐴} ⊆ {𝐴, 𝐵, 𝐶}
 
Theoremsnsstp2 3637 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐵} ⊆ {𝐴, 𝐵, 𝐶}
 
Theoremsnsstp3 3638 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐶} ⊆ {𝐴, 𝐵, 𝐶}
 
Theoremprsstp12 3639 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 3640 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 3641 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 3642 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 3643 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 3644 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
 
Theoremprsspwg 3645 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 3646 Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4085. (Contributed by Jim Kingdon, 10-Aug-2018.)
((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
 
Theoremsssnm 3647* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
(∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
 
Theoremeqsnm 3648* 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 3649 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
(((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
 
Theoremsstpr 3650 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 3651 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 3652 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 3653 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 3654 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 3655 Closed form of sneqr 3653. (Contributed by Scott Fenton, 1-Apr-2011.)
(𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
 
Theoremsneqbg 3656 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
(𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremsnsspw 3657 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
{𝐴} ⊆ 𝒫 𝐴
 
Theoremprsspw 3658 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 3659 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3661. (Contributed by Jim Kingdon, 21-Sep-2018.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
 
Theorempreqr2g 3660 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3662. (Contributed by Jim Kingdon, 21-Sep-2018.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵))
 
Theorempreqr1 3661 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 3662 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 3663 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 3664 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 3665 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 3666 Closed form of preq12b 3663. (Contributed by Scott Fenton, 28-Mar-2014.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
 
Theoremprneimg 3667 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 3668 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 3669 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
 
Theoremdfop 3670 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 3671 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 3672 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 3673 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
 
Theoremopeq1i 3674 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶
 
Theoremopeq2i 3675 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵
 
Theoremopeq12i 3676 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 3677 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
 
Theoremopeq2d 3678 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
 
Theoremopeq12d 3679 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 3680 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2 3681 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3 3682 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq1d 3683 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2d 3684 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3d 3685 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq123d 3686 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 3687 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵
 
Theoremnfopd 3688 Deduction version of bound-variable hypothesis builder nfop 3687. This shows how the deduction version of a not-free theorem such as nfop 3687 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)
 
Theoremopid 3689 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}
 
Theoremralunsn 3690* 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 3691* 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 3692 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 3693 Expansion of an ordered pair when the first member is a proper class. See also opprc 3692. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremopprc2 3694 Expansion of an ordered pair when the second member is a proper class. See also opprc 3692. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremoprcl 3695 If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorempwsnss 3696 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
{∅, {𝐴}} ⊆ 𝒫 {𝐴}
 
Theorempwpw0ss 3697 Compute the power set of the power set of the empty set. (See pw0 3633 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48 (but with subset in place of equality). (Contributed by Jim Kingdon, 12-Aug-2018.)
{∅, {∅}} ⊆ 𝒫 {∅}
 
Theorempwprss 3698 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵}
 
Theorempwtpss 3699 The power set of an unordered triple. (Contributed by Jim Kingdon, 13-Aug-2018.)
(({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}})) ⊆ 𝒫 {𝐴, 𝐵, 𝐶}
 
Theorempwpwpw0ss 3700 Compute the power set of the power set of the power set of the empty set. (See also pw0 3633 and pwpw0ss 3697.) (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}
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