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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopprc 4801 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 4802 Expansion of an ordered pair when the first member is a proper class. See also opprc 4801. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Theoremopprc2 4803 Expansion of an ordered pair when the second member is a proper class. See also opprc 4801. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Theoremoprcl 4804 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 4805 The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
𝒫 {𝐴} = {∅, {𝐴}}

TheorempwsnOLD 4806 Obsolete version of pwsn 4805 as of 14-Apr-2024. Note that the proof is essentially the same once one inlines sssn 4732 in the proof of pwsn 4805. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 {𝐴} = {∅, {𝐴}}

Theorempwpr 4807 The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}})

Theorempwtp 4808 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 4809 Compute the power set of the power set of the power set of the empty set. (See also pw0 4718 and pwpw0 4719.) (Contributed by NM, 2-May-2009.)
𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})

Theorempwv 4810 The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235.

The collection of all classes is of course larger than V, which is the collection of all sets. But 𝒫 V, being a class, cannot contain proper classes, so 𝒫 V is actually no larger than V. This fact is exploited in ncanth 7096. (Contributed by NM, 14-Sep-2003.)

𝒫 V = V

Theoremprproe 4811* 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 4812 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)

2.1.19  The union of a class

Syntaxcuni 4813 Extend class notation to include the union of a class. Read: "union (of) 𝐴".
class 𝐴

Definitiondf-uni 4814* 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 28209). This is similar to the union of two classes df-un 3913. (Contributed by NM, 23-Aug-1993.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥𝑦𝑦𝐴)}

Theoremdfuni2 4815* Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∃𝑦𝐴 𝑥𝑦}

Theoremeluni 4816* Membership in class union. (Contributed by NM, 22-May-1994.)
(𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))

Theoremeluni2 4817* Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
(𝐴 𝐵 ↔ ∃𝑥𝐵 𝐴𝑥)

Theoremelunii 4818 Membership in class union. (Contributed by NM, 24-Mar-1995.)
((𝐴𝐵𝐵𝐶) → 𝐴 𝐶)

Theoremnfunid 4819 Deduction version of nfuni 4820. (Contributed by NM, 18-Feb-2013.)
(𝜑𝑥𝐴)       (𝜑𝑥 𝐴)

Theoremnfuni 4820 Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴       𝑥 𝐴

Theoremuniss 4821 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)

Theoremunissi 4822 Subclass relationship for subclass union. Inference form of uniss 4821. (Contributed by David Moews, 1-May-2017.)
𝐴𝐵        𝐴 𝐵

Theoremunissd 4823 Subclass relationship for subclass union. Deduction form of uniss 4821. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 𝐴 𝐵)

Theoremunieq 4824 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.) (Proof shortened by BJ, 13-Apr-2024.)
(𝐴 = 𝐵 𝐴 = 𝐵)

TheoremunieqOLD 4825 Obsolete version of unieq 4824 as of 13-Apr-2024. (Contributed by NM, 10-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) 29-Jun-2011.)
(𝐴 = 𝐵 𝐴 = 𝐵)

Theoremunieqi 4826 Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵        𝐴 = 𝐵

Theoremunieqd 4827 Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)

Theoremeluniab 4828* 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 4829* Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
(𝐴 {𝑥𝐵𝜑} ↔ ∃𝑥𝐵 (𝐴𝑥𝜑))

Theoremunipr 4830 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 4831 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))

Theoremunisng 4832 A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremunisn 4833 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

Theoremunisn3 4834* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
(𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)

Theoremdfnfc2 4835* 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))

Theoremuniun 4836 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 4837 The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8364 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝐵) ⊆ ( 𝐴 𝐵)

Theoremssuni 4838 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)

Theoremuni0b 4839 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 4840* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)

Theoremuni0 4841 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Contributed by NM, 16-Sep-1993.) Remove use of ax-nul 5186. (Revised by Eric Schmidt, 4-Apr-2007.)
∅ = ∅

Theoremcsbuni 4842 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 4843 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 4844 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
(( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)

Theoremunissb 4845* 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 4846* 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 4948 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.)
(∀𝑥𝐴𝑦𝐵 𝑥𝑦 𝐴 𝐵)

Theoremunidif 4847* 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 4848* Relationship implying union. (Contributed by NM, 10-Nov-1999.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝑥𝐴) → 𝐴 = 𝐵)

Theoremunimax 4849* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
(𝐴𝐵 {𝑥𝐵𝑥𝐴} = 𝐴)

Theorempwuni 4850 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

2.1.20  The intersection of a class

Syntaxcint 4851 Extend class notation to include the intersection of a class. Read: "intersection (of) 𝐴".
class 𝐴

Definitiondf-int 4852* 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 3915. (Contributed by NM, 18-Aug-1993.)
𝐴 = {𝑥 ∣ ∀𝑦(𝑦𝐴𝑥𝑦)}

Theoremdfint2 4853* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
𝐴 = {𝑥 ∣ ∀𝑦𝐴 𝑥𝑦}

Theoreminteq 4854 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
(𝐴 = 𝐵 𝐴 = 𝐵)

Theoreminteqi 4855 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)
𝐴 = 𝐵        𝐴 = 𝐵

Theoreminteqd 4856 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐴 = 𝐵)       (𝜑 𝐴 = 𝐵)

Theoremelint 4857* Membership in class intersection. (Contributed by NM, 21-May-1994.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))

Theoremelint2 4858* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V       (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)

Theoremelintg 4859* 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))

Theoremelinti 4860 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴 𝐵 → (𝐶𝐵𝐴𝐶))

Theoremnfint 4861 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
𝑥𝐴       𝑥 𝐴

Theoremelintab 4862* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))

Theoremelintrab 4863* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
𝐴 ∈ V       (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))

Theoremelintrabg 4864* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)
(𝐴𝑉 → (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥)))

Theoremint0 4865 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

Theoremintss1 4866 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 4867* 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 4868* 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 4869* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
𝐴 {𝑥𝐵𝐴𝑥}

Theoremssmin 4870* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)
𝐴 {𝑥 ∣ (𝐴𝑥𝜑)}

Theoremintmin 4871* 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 4872 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐴𝐵 𝐵 𝐴)

Theoremintssuni 4873 The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
(𝐴 ≠ ∅ → 𝐴 𝐴)

Theoremssintrab 4874* Subclass of the intersection of a restricted class abstraction. (Contributed by NM, 30-Jan-2015.)
(𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))

Theoremunissint 4875 If the union of a class is included in its intersection, the class is either the empty set or a singleton (uniintsn 4888). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( 𝐴 𝐴 ↔ (𝐴 = ∅ ∨ 𝐴 = 𝐴))

Theoremintssuni2 4876 Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐴 𝐵)

Theoremintminss 4877* 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 4878* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V        {𝑥𝐴𝑥} = 𝐴

Theoremintmin3 4879* 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)

Theoremintmin4 4880* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
(𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})

Theoremintab 4881* The intersection of a special case of a class abstraction. 𝑦 may be free in 𝜑 and 𝐴, which can be thought of a 𝜑(𝑦) and 𝐴(𝑦). Typically, abrexex2 7656 or abexssex 7657 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝐴 ∈ V    &   {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)} ∈ V        {𝑥 ∣ ∀𝑦(𝜑𝐴𝑥)} = {𝑥 ∣ ∃𝑦(𝜑𝑥 = 𝐴)}

Theoremint0el 4882 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
(∅ ∈ 𝐴 𝐴 = ∅)

Theoremintun 4883 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
(𝐴𝐵) = ( 𝐴 𝐵)

Theoremintpr 4884 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
𝐴 ∈ V    &   𝐵 ∈ V        {𝐴, 𝐵} = (𝐴𝐵)

Theoremintprg 4885 The intersection of a pair is the intersection of its members. Closed form of intpr 4884. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Theoremintsng 4886 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremintsn 4887 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)
𝐴 ∈ V        {𝐴} = 𝐴

Theoremuniintsn 4888* Two ways to express "𝐴 is a singleton." See also en1 8563, en1b 8564, card1 9385, and eusn 4640. (Contributed by NM, 2-Aug-2010.)
( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremuniintab 4889 The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of 𝜑(𝑥). (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 {𝑥𝜑} = {𝑥𝜑})

Theoremintunsn 4890 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
𝐵 ∈ V        (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)

Theoremrint0 4891 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑋) = 𝐴)

Theoremelrint 4892* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))

Theoremelrint2 4893* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))

2.1.21  Indexed union and intersection

Syntaxciun 4894 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation 𝑥𝐴𝐵, with the same union symbol as cuni 4813. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class 𝑥𝐴 𝐵

Syntaxciin 4895 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation 𝑥𝐴𝐵, with the same intersection symbol as cint 4851. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.
class 𝑥𝐴 𝐵

Definitiondf-iun 4896* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, 𝐴 is independent of 𝑥 (although this is not required by the definition), and 𝐵 depends on 𝑥 i.e. can be read informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the index set, and 𝐵 the indexed set. In most books, 𝑥𝐴 is written as a subscript or underneath a union symbol . We use a special union symbol to make it easier to distinguish from plain class union. In many theorems, you will see that 𝑥 and 𝐴 are in the same distinct variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a distinct variable group (meaning that can be thought of as 𝐵(𝑥) i.e. can be substituted with a class expression containing 𝑥). An alternate definition tying indexed union to ordinary union is dfiun2 4933. Theorem uniiun 4957 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6989 and funiunfv 6990 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}

Definitiondf-iin 4897* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4896. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4934. Theorem intiin 4958 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}

Theoremeliun 4898* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(𝐴 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝐴𝐶)

Theoremeliin 4899* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremeliuni 4900* Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
(𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)

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