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Theorem List for Metamath Proof Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremintminss 4501* 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 4502* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V        {𝑥𝐴𝑥} = 𝐴

Theoremintmin3 4503* 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 4504* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)
(𝐴 {𝑥𝜑} → {𝑥 ∣ (𝐴𝑥𝜑)} = {𝑥𝜑})

Theoremintab 4505* 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 7145 or abexssex 7146 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 4506 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
(∅ ∈ 𝐴 𝐴 = ∅)

Theoremintun 4507 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 4508 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 4509 The intersection of a pair is the intersection of its members. Closed form of intpr 4508. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Theoremintsng 4510 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 {𝐴} = 𝐴)

Theoremintsn 4511 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 4512* Two ways to express "𝐴 is a singleton." See also en1 8020, en1b 8021, card1 8791, and eusn 4263. (Contributed by NM, 2-Aug-2010.)
( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremuniintab 4513 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 4514 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)
𝐵 ∈ V        (𝐴 ∪ {𝐵}) = ( 𝐴𝐵)

Theoremrint0 4515 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑋) = 𝐴)

Theoremelrint 4516* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 ∈ (𝐴 𝐵) ↔ (𝑋𝐴 ∧ ∀𝑦𝐵 𝑋𝑦))

Theoremelrint2 4517* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋𝐴 → (𝑋 ∈ (𝐴 𝐵) ↔ ∀𝑦𝐵 𝑋𝑦))

2.1.20  Indexed union and intersection

Syntaxciun 4518 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 4434. 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 4519 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 4473. 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 4520* 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 4552. Theorem uniiun 4571 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6502 and funiunfv 6503 are useful when 𝐵 is a function. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}

Definitiondf-iin 4521* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4520. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4553. Theorem intiin 4572 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}

Theoremeliun 4522* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(𝐴 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝐴𝐶)

Theoremeliin 4523* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(𝐴𝑉 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))

Theoremeliuni 4524* Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
(𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐸𝐶) → 𝐸 𝑥𝐷 𝐵)

Theoremiuncom 4525* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 𝐶 = 𝑦𝐵 𝑥𝐴 𝐶

Theoremiuncom4 4526 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremiunconst 4527* Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)

Theoremiinconst 4528* Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝐵)

Theoremiuniin 4529* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶

Theoremiunss1 4530* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝐵 𝑥𝐴 𝐶 𝑥𝐵 𝐶)

Theoremiinss1 4531* Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
(𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)

Theoremiuneq1 4532* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiineq1 4533* Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremss2iun 4534 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Theoremiuneq2 4535 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiineq2 4536 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq2i 4537 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
(𝑥𝐴𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Theoremiineq2i 4538 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
(𝑥𝐴𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Theoremiineq2d 4539 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq2dv 4540* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiineq2dv 4541* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq12df 4542 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiuneq1d 4543* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiuneq12d 4544* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiuneq2d 4545* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremnfiun 4546 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiin 4547 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiu1 4548 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
𝑥 𝑥𝐴 𝐵

Theoremnfii1 4549 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
𝑥 𝑥𝐴 𝐵

Theoremdfiun2g 4550* Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremdfiin2g 4551* Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremdfiun2 4552* Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
𝐵 ∈ V        𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremdfiin2 4553* Alternate definition of indexed intersection when 𝐵 is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐵 ∈ V        𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremdfiunv2 4554* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}

Theoremcbviun 4555* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviin 4556* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviunv 4557* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviinv 4558* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremiunss 4559* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)

Theoremssiun 4560* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐶𝐵𝐶 𝑥𝐴 𝐵)

Theoremssiun2 4561 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝑥𝐴𝐵 𝑥𝐴 𝐵)

Theoremssiun2s 4562* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Theoremiunss2 4563* A subclass condition on the members of two indexed classes 𝐶(𝑥) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4468. (Contributed by NM, 9-Dec-2004.)
(∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)

Theoremiunab 4564* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}

Theoremiunrab 4565* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}

Theoremiunxdif2 4566* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(𝑥 = 𝑦𝐶 = 𝐷)       (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)

Theoremssiinf 4567 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
𝑥𝐶       (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)

Theoremssiin 4568* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)

Theoremiinss 4569* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)

Theoremiinss2 4570 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(𝑥𝐴 𝑥𝐴 𝐵𝐵)

Theoremuniiun 4571* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥

Theoremintiin 4572* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥

Theoremiunid 4573* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
𝑥𝐴 {𝑥} = 𝐴

Theoremiun0 4574 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴 ∅ = ∅

Theorem0iun 4575 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥 ∈ ∅ 𝐴 = ∅

Theorem0iin 4576 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝐴 = V

Theoremviin 4577* Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2068 and abid2 2744. When 𝐴 = 𝑥, this evaluates to by intiin 4572 and intv 4839. (Contributed by NM, 11-Sep-2008.)
𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}

Theoremiunn0 4578* There is a nonempty class in an indexed collection 𝐵(𝑥) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 𝐵 ≠ ∅)

Theoremiinab 4579* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}

Theoremiinrab 4580* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
(𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})

Theoremiinrab2 4581* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
( 𝑥𝐴 {𝑦𝐵𝜑} ∩ 𝐵) = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑}

Theoremiunin2 4582* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4571 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)

Theoremiunin1 4583* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4571 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)

Theoremiinun2 4584* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4572 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)

Theoremiundif2 4585* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4572 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)

Theorem2iunin 4586* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)

Theoremiindif2 4587* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4571 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))

Theoremiinin2 4588* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4572 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))

Theoremiinin1 4589* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4572 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))

Theoremiinvdif 4590* The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)

Theoremelriin 4591* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))

Theoremriin0 4592* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)

Theoremriinn0 4593* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)

Theoremriinrab 4594* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}

Theoremsymdif0 4595 Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ ∅) = 𝐴

Theoremsymdifv 4596 Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ V) = (V ∖ 𝐴)

Theoremsymdifid 4597 Symmetric difference with self yields the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
(𝐴𝐴) = ∅

Theoremiinxsng 4598* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Theoremiinxprg 4599* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))

Theoremiunxsng 4600* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)

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