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Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopeq2 4401 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 4402 Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.)
((𝐴 = 𝐶𝐵 = 𝐷) → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
 
Theoremopeq1i 4403 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐴, 𝐶⟩ = ⟨𝐵, 𝐶
 
Theoremopeq2i 4404 Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
𝐴 = 𝐵       𝐶, 𝐴⟩ = ⟨𝐶, 𝐵
 
Theoremopeq12i 4405 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 4406 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
 
Theoremopeq2d 4407 Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
 
Theoremopeq12d 4408 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 4409 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2 4410 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3 4411 Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
(𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq1d 4412 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 
Theoremoteq2d 4413 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 
Theoremoteq3d 4414 Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
 
Theoremoteq123d 4415 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 4416 Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴, 𝐵
 
Theoremnfopd 4417 Deduction version of bound-variable hypothesis builder nfop 4416. This shows how the deduction version of a not-free theorem such as nfop 4416 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥𝐴, 𝐵⟩)
 
Theoremcsbopg 4418 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⟩)
 
Theoremopid 4419 The ordered pair 𝐴, 𝐴 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) (Avoid depending on this detail.)
𝐴 ∈ V       𝐴, 𝐴⟩ = {{𝐴}}
 
Theoremralunsn 4420* 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 4421* 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 4422 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 4423 Expansion of an ordered pair when the first member is a proper class. See also opprc 4422. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremopprc2 4424 Expansion of an ordered pair when the second member is a proper class. See also opprc 4422. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
 
Theoremoprcl 4425 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))
 
Theorempwsn 4426 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
𝒫 {𝐴} = {∅, {𝐴}}
 
TheorempwsnALT 4427 Alternate proof of pwsn 4426, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 {𝐴} = {∅, {𝐴}}
 
Theorempwpr 4428 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}})
 
Theorempwtp 4429 The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
𝒫 {𝐴, 𝐵, 𝐶} = (({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ∪ ({{𝐶}, {𝐴, 𝐶}} ∪ {{𝐵, 𝐶}, {𝐴, 𝐵, 𝐶}}))
 
Theorempwpwpw0 4430 Compute the power set of the power set of the power set of the empty set. (See also pw0 4341 and pwpw0 4342.) (Contributed by NM, 2-May-2009.)
𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
 
Theorempwv 4431 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
 
Theoremprproe 4432* For an element of a proper unordered pair of elements of a class 𝑉, there is another (different) element of the class 𝑉 which is an element of the proper pair. (Contributed by AV, 18-Dec-2021.)
((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴𝐵 ∧ (𝐴𝑉𝐵𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵})
 
Theorem3elpr2eq 4433 If there are three elements in a proper unordered pair, and two of them are different from the third one, the two must be equal. (Contributed by AV, 19-Dec-2021.)
(((𝑋 ∈ {𝐴, 𝐵} ∧ 𝑌 ∈ {𝐴, 𝐵} ∧ 𝑍 ∈ {𝐴, 𝐵}) ∧ (𝑌𝑋𝑍𝑋)) → 𝑌 = 𝑍)
 
2.1.18  The union of a class
 
Syntaxcuni 4434 Extend class notation to include the union of a class (read: 'union 𝐴')
class 𝐴
 
Definitiondf-uni 4435* 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} (ex-uni 27267). This is similar to the union of two classes df-un 3577. (Contributed by NM, 23-Aug-1993.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}
 
Theoremdfuni2 4436* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}
 
Theoremeluni 4437* Membership in class union. (Contributed by NM, 22-May-1994.)
(𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 
Theoremeluni2 4438* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
(𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)
 
Theoremelunii 4439 Membership in class union. (Contributed by NM, 24-Mar-1995.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
Theoremnfuni 4440 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴       𝑥 𝐴
 
Theoremnfunid 4441 Deduction version of nfuni 4440. (Contributed by NM, 18-Feb-2013.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)
 
Theoremunieq 4442 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 4443 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵        𝐴 = 𝐵
 
Theoremunieqd 4444 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)
 
Theoremeluniab 4445* 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 4446* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
(𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))
 
Theoremunipr 4447 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 4448 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 4449 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 4450 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
(𝐴𝑉 {𝐴} = 𝐴)
 
Theoremunisn3 4451* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
(𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
 
Theoremdfnfc2 4452* An alternative statement of the effective freeness of a class 𝐴, when it is a set. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof shortened by JJ, 26-Jul-2021.)
(∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
 
Theoremdfnfc2OLD 4453* Obsolete proof of dfnfc2 4452 as of 26-Jul-2021. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝐴𝑉 → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
 
Theoremuniun 4454 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 4455 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7824 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝐵) ⊆ ( 𝐴 𝐵)
 
Theoremuniss 4456 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 4457 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by JJ, 26-Jul-2021.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
TheoremssuniOLD 4458 Obsolete proof of ssuni 4457 as of 26-Jul-2021. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)
 
Theoremunissi 4459 Subclass relationship for subclass union. Inference form of uniss 4456. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵        𝐴 𝐵
 
Theoremunissd 4460 Subclass relationship for subclass union. Deduction form of uniss 4456. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 𝐴 𝐵)
 
Theoremuni0b 4461 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 4462* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
 
Theoremuni0 4463 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul 4787 by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
∅ = ∅
 
Theoremcsbuni 4464 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 22-Aug-2018.)
𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵
 
Theoremelssuni 4465 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 4466 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
(( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
 
Theoremunissb 4467* 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 4468* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. See iunss2 4563 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)
 
Theoremunidif 4469* 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 4470* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)
 
Theoremunimax 4471* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
(𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)
 
Theorempwuni 4472 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
𝐴 ⊆ 𝒫 𝐴
 
2.1.19  The intersection of a class
 
Syntaxcint 4473 Extend class notation to include the intersection of a class (read: 'intersect 𝐴').
class 𝐴
 
Definitiondf-int 4474* 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 3579. (Contributed by NM, 18-Aug-1993.)
𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}
 
Theoremdfint2 4475* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}
 
Theoreminteq 4476 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
(𝐴 = 𝐵 𝐴 = 𝐵)
 
Theoreminteqi 4477 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
𝐴 = 𝐵        𝐴 = 𝐵
 
Theoreminteqd 4478 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)
 
Theoremelint 4479* Membership in class intersection. (Contributed by NM, 21-May-1994.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
 
Theoremelint2 4480* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
 
Theoremelintg 4481* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.)
(𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
 
TheoremelintgOLD 4482* Obsolete proof of elintg 4481 as of 26-Jul-2021. (Contributed by NM, 20-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
 
Theoremelinti 4483 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
 
Theoremnfint 4484 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
𝑥𝐴       𝑥 𝐴
 
Theoremelintab 4485* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
 
Theoremelintrab 4486* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ V       (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
 
Theoremelintrabg 4487* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
(𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))
 
Theoremint0 4488 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)
∅ = V
 
Theoremint0OLD 4489 Obsolete proof of int0 4488 as of 26-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
∅ = V
 
Theoremintss1 4490 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 4491* 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 4492* 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 4493* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
𝐴 {𝑥𝐵𝐴𝑥}
 
Theoremssmin 4494* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}
 
Theoremintmin 4495* 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 4496 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐴𝐵 𝐵 𝐴)
 
Theoremintssuni 4497 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(𝐴 ≠ ∅ → 𝐴 𝐴)
 
Theoremssintrab 4498* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)
(𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
 
Theoremunissint 4499 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4512). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))
 
Theoremintssuni2 4500 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)
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