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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiunss1 4901* 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 4902* Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
(𝐴𝐵 𝑥𝐵 𝐶 𝑥𝐴 𝐶)

Theoremiuneq1 4903* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiineq1 4904* Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐵 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremss2iun 4905 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 4906 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiineq2 4907 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 4908 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
(𝑥𝐴𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Theoremiineq2i 4909 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
(𝑥𝐴𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Theoremiineq2d 4910 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq2dv 4911* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiineq2dv 4912* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremiuneq12df 4913 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 4914* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremiuneq12d 4915* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)

Theoremiuneq2d 4916* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
(𝜑𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremnfiun 4917* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2380. See nfiung 4919 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiin 4918* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2380. See nfiing 4920 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiung 4919 Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2380. See nfiun 4917 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiing 4920 Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2380. See nfiin 4918 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵

Theoremnfiu1 4921 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
𝑥 𝑥𝐴 𝐵

Theoremnfii1 4922 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
𝑥 𝑥𝐴 𝐵

Theoremdfiun2g 4923* 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.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremdfiun2gOLD 4924* Obsolete proof of dfiun2g 4923 as of 11-Aug-2023. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremdfiin2g 4925* 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 4926* 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 4927* 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 4928* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
𝑥𝐴 𝑦𝐵 𝐶 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧𝐶}

Theoremcbviun 4929* 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.) Add disjoint variable condition to avoid ax-13 2380. See cbviung 4931 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviin 4930* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2380. See cbviing 4932 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviung 4931* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2380. See cbviun 4929 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviing 4932* Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2380. See cbviin 4930 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviunv 4933* 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.) Add disjoint variable condition to avoid ax-13 2380. See cbviunvg 4935 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviinv 4934* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2380. See cbviinvg 4936 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviunvg 4935* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2380. Usage of the weaker cbviunv 4933 is preferred. (Contributed by NM, 15-Sep-2003.) (New usage is discouraged.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremcbviinvg 4936* Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2380. Usage of the weaker cbviinv 4934 is preferred. (Contributed by Jeff Hankins, 26-Aug-2009.) (New usage is discouraged.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶

Theoremiunssf 4937 Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐶       ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)

Theoremiunss 4938* 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 4939* 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 4940 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 4941* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Theoremiunss2 4942* 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 4837. (Contributed by NM, 9-Dec-2004.)
(∀𝑥𝐴𝑦𝐵 𝐶𝐷 𝑥𝐴 𝐶 𝑦𝐵 𝐷)

Theoremiunssd 4943* Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)

Theoremiunab 4944* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∃𝑥𝐴 𝜑}

Theoremiunrab 4945* 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 4946* 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 4947 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 4948* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)

Theoremiinss 4949* 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 4950 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(𝑥𝐴 𝑥𝐴 𝐵𝐵)

Theoremuniiun 4951* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥

Theoremintiin 4952* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥

Theoremiunid 4953* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
𝑥𝐴 {𝑥} = 𝐴

Theoremiun0 4954 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 4955 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥 ∈ ∅ 𝐴 = ∅

Theorem0iin 4956 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝐴 = V

Theoremviin 4957* Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2201 and abid2 2895. When 𝐴 = 𝑥, this evaluates to by intiin 4952 and intv 5237. (Contributed by NM, 11-Sep-2008.)
𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}

Theoremiunsn 4958* Indexed union of a singleton. Compare dfiun2 4926 and rnmpt 5802. (Contributed by Steven Nguyen, 7-Jun-2023.)
𝑥𝐴 {𝐵} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremiunn0 4959* 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 4960* Indexed intersection of a class abstraction. (Contributed by NM, 6-Dec-2011.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}

Theoremiinrab 4961* Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
(𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})

Theoremiinrab2 4962* Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
( 𝑥𝐴 {𝑦𝐵𝜑} ∩ 𝐵) = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑}

Theoremiunin2 4963* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4951 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)

Theoremiunin1 4964* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4951 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)

Theoremiinun2 4965* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4952 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)

Theoremiundif2 4966* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4952 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)

Theoremiindif1 4967* Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶))

Theorem2iunin 4968* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)

Theoremiindif2 4969* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4951 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))

Theoremiinin2 4970* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4952 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))

Theoremiinin1 4971* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4952 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))

Theoremiinvdif 4972* The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)

Theoremelriin 4973* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))

Theoremriin0 4974* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)

Theoremriinn0 4975* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)

Theoremriinrab 4976* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}

Theoremsymdif0 4977 Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ ∅) = 𝐴

Theoremsymdifv 4978 The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ V) = (V ∖ 𝐴)

Theoremsymdifid 4979 The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
(𝐴𝐴) = ∅

Theoremiinxsng 4980* 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 4981* 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 4982* 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)

Theoremiunxsn 4983* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶

Theoremiunxsngf 4984* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.) Avoid ax-13 2380. (Revised by Gino Giotto, 19-May-2023.)
𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Theoremiunun 4985 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)

Theoremiunxun 4986 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)

Theoremiunxdif3 4987* An indexed union where some terms are the empty set. See iunxdif2 4946. (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐸       (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)

Theoremiunxprg 4988* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))

Theoremiunxiun 4989* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶

Theoremiinuni 4990* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)

Theoremiununi 4991* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))

Theoremsspwuni 4992 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)

Theorempwssb 4993* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)

Theoremelpwpw 4994 Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)
(𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴𝐵))

Theorempwpwab 4995* The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)
𝒫 𝒫 𝐴 = {𝑥 𝑥𝐴}

Theorempwpwssunieq 4996* The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)
{𝑥 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Theoremelpwuni 4997 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))

Theoremiinpw 4998* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥

Theoremiunpwss 4999* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴

Theoremintss2 5000 A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.)
(𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → 𝐴𝑋))

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