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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopthpr 3601 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 3602 Closed form of preq12b 3599. (Contributed by Scott Fenton, 28-Mar-2014.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
 
Theoremprneimg 3603 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 3604 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 3605 Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
 
Theoremdfop 3606 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 3607 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 3608 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 3609 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
 
Theoremopeq1i 3610 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶
 
Theoremopeq2i 3611 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵
 
Theoremopeq12i 3612 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 3613 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
 
Theoremopeq2d 3614 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
 
Theoremopeq12d 3615 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 3616 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2 3617 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3 3618 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq1d 3619 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2d 3620 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3d 3621 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq123d 3622 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 3623 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵
 
Theoremnfopd 3624 Deduction version of bound-variable hypothesis builder nfop 3623. This shows how the deduction version of a not-free theorem such as nfop 3623 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)
 
Theoremopid 3625 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}
 
Theoremralunsn 3626* 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 3627* 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 3628 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 3629 Expansion of an ordered pair when the first member is a proper class. See also opprc 3628. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremopprc2 3630 Expansion of an ordered pair when the second member is a proper class. See also opprc 3628. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremoprcl 3631 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 3632 The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
{∅, {𝐴}} ⊆ 𝒫 {𝐴}
 
Theorempwpw0ss 3633 Compute the power set of the power set of the empty set. (See pw0 3569 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 3634 The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵}
 
Theorempwtpss 3635 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 3636 Compute the power set of the power set of the power set of the empty set. (See also pw0 3569 and pwpw0ss 3633.) (Contributed by Jim Kingdon, 13-Aug-2018.)
({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}
 
Theorempwv 3637 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 3638 Extend class notation to include the union of a class (read: 'union 𝐴')
class 𝐴
 
Definitiondf-uni 3639* 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 2992. (Contributed by NM, 23-Aug-1993.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
 
Theoremdfuni2 3640* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
 
Theoremeluni 3641* Membership in class union. (Contributed by NM, 22-May-1994.)
(𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 
Theoremeluni2 3642* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
(𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
 
Theoremelunii 3643 Membership in class union. (Contributed by NM, 24-Mar-1995.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
Theoremnfuni 3644 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴       𝑥 𝐴
 
Theoremnfunid 3645 Deduction version of nfuni 3644. (Contributed by NM, 18-Feb-2013.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
Theoremcsbunig 3646 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremunieq 3647 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 3648 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵        𝐴 = 𝐵
 
Theoremunieqd 3649 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)
 
Theoremeluniab 3650* 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 3651* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
(𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
 
Theoremunipr 3652 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 3653 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 3654 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 3655 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 3656* An alternate 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 3657 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 3658 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 3659 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 3660 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
Theoremunissi 3661 Subclass relationship for subclass union. Inference form of uniss 3659. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵        𝐴 𝐵
 
Theoremunissd 3662 Subclass relationship for subclass union. Deduction form of uniss 3659. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 𝐴 𝐵)
 
Theoremuni0b 3663 The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)
( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
 
Theoremuni0c 3664* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
 
Theoremuni0 3665 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
∅ = ∅
 
Theoremelssuni 3666 An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)
(𝐴𝐵𝐴 𝐵)
 
Theoremunissel 3667 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
(( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
 
Theoremunissb 3668* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)
( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremuniss2 3669* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
 
Theoremunidif 3670* If the difference 𝐴𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝑥𝑦 (𝐴𝐵) = 𝐴)
 
Theoremssunieq 3671* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
 
Theoremunimax 3672* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
(𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
 
2.1.19  The intersection of a class
 
Syntaxcint 3673 Extend class notation to include the intersection of a class (read: 'intersect 𝐴').
class 𝐴
 
Definitiondf-int 3674* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 } . Compare this with the intersection of two classes, df-in 2994. (Contributed by NM, 18-Aug-1993.)
𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
 
Theoremdfint2 3675* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
 
Theoreminteq 3676 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
(𝐴 = 𝐵 𝐴 = 𝐵)
 
Theoreminteqi 3677 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
𝐴 = 𝐵        𝐴 = 𝐵
 
Theoreminteqd 3678 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)
 
Theoremelint 3679* Membership in class intersection. (Contributed by NM, 21-May-1994.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
 
Theoremelint2 3680* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
 
Theoremelintg 3681* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
(𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
 
Theoremelinti 3682 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
 
Theoremnfint 3683 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
𝑥𝐴       𝑥 𝐴
 
Theoremelintab 3684* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
 
Theoremelintrab 3685* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ V       (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
 
Theoremelintrabg 3686* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
(𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
 
Theoremint0 3687 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
∅ = V
 
Theoremintss1 3688 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
(𝐴𝐵 𝐵𝐴)
 
Theoremssint 3689* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
(𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
 
Theoremssintab 3690* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
 
Theoremssintub 3691* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
𝐴 {𝑥𝐵𝐴𝑥}
 
Theoremssmin 3692* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
 
Theoremintmin 3693* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
 
Theoremintss 3694 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 𝐵 𝐴)
 
Theoremintssunim 3695* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(∃𝑥 𝑥𝐴 𝐴 𝐴)
 
Theoremssintrab 3696* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
(𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
 
Theoremintssuni2m 3697* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → 𝐴 𝐵)
 
Theoremintminss 3698* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → {𝑥𝐵𝜑} ⊆ 𝐴)
 
Theoremintmin2 3699* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V        {𝑥𝐴𝑥} = 𝐴
 
Theoremintmin3 3700* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   𝜓       (𝐴𝑉 {𝑥𝜑} ⊆ 𝐴)
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