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Theorem List for Intuitionistic Logic Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdifprsn2 3501 Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
(𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
 
Theoremdiftpsn3 3502 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 3503 (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 3498. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
 
Theoremdifsnpssim 3504 (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if 𝐴 is a member of 𝐵. In classical logic, the converse holds as well. (Contributed by Jim Kingdon, 9-Aug-2018.)
(𝐴𝐵 → (𝐵 ∖ {𝐴}) ⊊ 𝐵)
 
Theoremsnssi 3505 The singleton of an element of a class is a subset of the class. (Contributed by NM, 6-Jun-1994.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)
 
Theoremsnssd 3506 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 3507 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 6067. (Contributed by Jim Kingdon, 10-Aug-2018.)
(𝐵𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴)
 
Theorempw0 3508 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 3509 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)
{𝐴} ⊆ {𝐴, 𝐵}
 
Theoremsnsspr2 3510 A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)
{𝐵} ⊆ {𝐴, 𝐵}
 
Theoremsnsstp1 3511 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐴} ⊆ {𝐴, 𝐵, 𝐶}
 
Theoremsnsstp2 3512 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐵} ⊆ {𝐴, 𝐵, 𝐶}
 
Theoremsnsstp3 3513 A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
{𝐶} ⊆ {𝐴, 𝐵, 𝐶}
 
Theoremprsstp12 3514 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 3515 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 3516 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 3517 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 3518 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 3519 A pair of elements of a class is a subset of the class. (Contributed by NM, 16-Jan-2015.)
((𝐴𝐶𝐵𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
 
Theoremprsspwg 3520 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 3521 Empty set and the singleton itself are subsets of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
 
Theoremsssnm 3522* The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
(∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
 
Theoremeqsnm 3523* 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 3524 The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
(((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
 
Theoremsstpr 3525 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 3526 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 3527 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 3528 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 3529 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 3530 Closed form of sneqr 3528. (Contributed by Scott Fenton, 1-Apr-2011.)
(𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
 
Theoremsneqbg 3531 Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
(𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
 
Theoremsnsspw 3532 The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
{𝐴} ⊆ 𝒫 𝐴
 
Theoremprsspw 3533 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 3534 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3536. (Contributed by Jim Kingdon, 21-Sep-2018.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐶} = {𝐵, 𝐶} → 𝐴 = 𝐵))
 
Theorempreqr2g 3535 Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3537. (Contributed by Jim Kingdon, 21-Sep-2018.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐶, 𝐴} = {𝐶, 𝐵} → 𝐴 = 𝐵))
 
Theorempreqr1 3536 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 3537 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 3538 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 3539 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 3540 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 3541 Closed form of preq12b 3538. (Contributed by Scott Fenton, 28-Mar-2014.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
 
Theoremprneimg 3542 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 3543 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 3544 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
 
Theoremdfop 3545 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 3546 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 3547 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 3548 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
 
Theoremopeq1i 3549 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶
 
Theoremopeq2i 3550 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵
 
Theoremopeq12i 3551 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 3552 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
 
Theoremopeq2d 3553 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
 
Theoremopeq12d 3554 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 3555 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2 3556 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3 3557 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq1d 3558 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2d 3559 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3d 3560 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq123d 3561 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 3562 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵
 
Theoremnfopd 3563 Deduction version of bound-variable hypothesis builder nfop 3562. This shows how the deduction version of a not-free theorem such as nfop 3562 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)
 
Theoremopid 3564 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}
 
Theoremralunsn 3565* 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 3566* 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 3567 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 3568 Expansion of an ordered pair when the first member is a proper class. See also opprc 3567. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremopprc2 3569 Expansion of an ordered pair when the second member is a proper class. See also opprc 3567. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremoprcl 3570 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 3571 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
{∅, {𝐴}} ⊆ 𝒫 {𝐴}
 
Theorempwpw0ss 3572 Compute the power set of the power set of the empty set. (See pw0 3508 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 3573 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵}
 
Theorempwtpss 3574 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 3575 Compute the power set of the power set of the power set of the empty set. (See also pw0 3508 and pwpw0ss 3572.) (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}
 
Theorempwv 3576 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
𝒫 V = V
 
2.1.18  The union of a class
 
Syntaxcuni 3577 Extend class notation to include the union of a class (read: 'union 𝐴')
class 𝐴
 
Definitiondf-uni 3578* Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } . This is similar to the union of two classes df-un 2919. (Contributed by NM, 23-Aug-1993.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
 
Theoremdfuni2 3579* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
 
Theoremeluni 3580* Membership in class union. (Contributed by NM, 22-May-1994.)
(𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 
Theoremeluni2 3581* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
(𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
 
Theoremelunii 3582 Membership in class union. (Contributed by NM, 24-Mar-1995.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
Theoremnfuni 3583 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴       𝑥 𝐴
 
Theoremnfunid 3584 Deduction version of nfuni 3583. (Contributed by NM, 18-Feb-2013.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
Theoremcsbunig 3585 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremunieq 3586 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴 = 𝐵 𝐴 = 𝐵)
 
Theoremunieqi 3587 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵        𝐴 = 𝐵
 
Theoremunieqd 3588 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)
 
Theoremeluniab 3589* Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
(𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
 
Theoremelunirab 3590* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
(𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
 
Theoremunipr 3591 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
𝐴 ∈ V    &   𝐵 ∈ V        {𝐴, 𝐵} = (𝐴𝐵)
 
Theoremuniprg 3592 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
 
Theoremunisn 3593 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V        {𝐴} = 𝐴
 
Theoremunisng 3594 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
(𝐴𝑉 {𝐴} = 𝐴)
 
Theoremdfnfc2 3595* An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.)
(∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
 
Theoremuniun 3596 The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
(𝐴𝐵) = ( 𝐴 𝐵)
 
Theoremuniin 3597 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝐵) ⊆ ( 𝐴 𝐵)
 
Theoremuniss 3598 Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝐵 𝐴 𝐵)
 
Theoremssuni 3599 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
Theoremunissi 3600 Subclass relationship for subclass union. Inference form of uniss 3598. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵        𝐴 𝐵
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