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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-iin 5001* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 5000. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 5038. Theorem intiin 5063 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)
∩ π‘₯ ∈ 𝐴 𝐡 = {𝑦 ∣ βˆ€π‘₯ ∈ 𝐴 𝑦 ∈ 𝐡}
 
Theoremeliun 5002* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)
(𝐴 ∈ βˆͺ π‘₯ ∈ 𝐡 𝐢 ↔ βˆƒπ‘₯ ∈ 𝐡 𝐴 ∈ 𝐢)
 
Theoremeliin 5003* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ ∩ π‘₯ ∈ 𝐡 𝐢 ↔ βˆ€π‘₯ ∈ 𝐡 𝐴 ∈ 𝐢))
 
Theoremeliuni 5004* Membership in an indexed union, one way. (Contributed by JJ, 27-Jul-2021.)
(π‘₯ = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   ((𝐴 ∈ 𝐷 ∧ 𝐸 ∈ 𝐢) β†’ 𝐸 ∈ βˆͺ π‘₯ ∈ 𝐷 𝐡)
 
Theoremiuncom 5005* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
βˆͺ π‘₯ ∈ 𝐴 βˆͺ 𝑦 ∈ 𝐡 𝐢 = βˆͺ 𝑦 ∈ 𝐡 βˆͺ π‘₯ ∈ 𝐴 𝐢
 
Theoremiuncom4 5006 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)
βˆͺ π‘₯ ∈ 𝐴 βˆͺ 𝐡 = βˆͺ βˆͺ π‘₯ ∈ 𝐴 𝐡
 
Theoremiunconst 5007* Indexed union of a constant class, i.e. where 𝐡 does not depend on π‘₯. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 β‰  βˆ… β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = 𝐡)
 
Theoremiinconst 5008* Indexed intersection of a constant class, i.e. where 𝐡 does not depend on π‘₯. (Contributed by Mario Carneiro, 6-Feb-2015.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 𝐡 = 𝐡)
 
Theoremiuneqconst 5009* Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)
(π‘₯ = 𝑋 β†’ 𝐡 = 𝐢)    β‡’   ((𝑋 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 = 𝐢) β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = 𝐢)
 
Theoremiuniin 5010* 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.)
βˆͺ π‘₯ ∈ 𝐴 ∩ 𝑦 ∈ 𝐡 𝐢 βŠ† ∩ 𝑦 ∈ 𝐡 βˆͺ π‘₯ ∈ 𝐴 𝐢
 
Theoremiinssiun 5011* An indexed intersection is a subset of the corresponding indexed union. (Contributed by Thierry Arnoux, 31-Dec-2021.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
Theoremiunss1 5012* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 βŠ† 𝐡 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 βŠ† βˆͺ π‘₯ ∈ 𝐡 𝐢)
 
Theoremiinss1 5013* Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)
(𝐴 βŠ† 𝐡 β†’ ∩ π‘₯ ∈ 𝐡 𝐢 βŠ† ∩ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiuneq1 5014* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐡 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 = βˆͺ π‘₯ ∈ 𝐡 𝐢)
 
Theoremiineq1 5015* Equality theorem for indexed intersection. (Contributed by NM, 27-Jun-1998.)
(𝐴 = 𝐡 β†’ ∩ π‘₯ ∈ 𝐴 𝐢 = ∩ π‘₯ ∈ 𝐡 𝐢)
 
Theoremss2iun 5016 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(βˆ€π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiuneq2 5017 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)
(βˆ€π‘₯ ∈ 𝐴 𝐡 = 𝐢 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiineq2 5018 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(βˆ€π‘₯ ∈ 𝐴 𝐡 = 𝐢 β†’ ∩ π‘₯ ∈ 𝐴 𝐡 = ∩ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiuneq2i 5019 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
(π‘₯ ∈ 𝐴 β†’ 𝐡 = 𝐢)    β‡’   βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ π‘₯ ∈ 𝐴 𝐢
 
Theoremiineq2i 5020 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
(π‘₯ ∈ 𝐴 β†’ 𝐡 = 𝐢)    β‡’   βˆ© π‘₯ ∈ 𝐴 𝐡 = ∩ π‘₯ ∈ 𝐴 𝐢
 
Theoremiineq2d 5021 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)
β„²π‘₯πœ‘    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ ∩ π‘₯ ∈ 𝐴 𝐡 = ∩ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiuneq2dv 5022* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiineq2dv 5023* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ ∩ π‘₯ ∈ 𝐴 𝐡 = ∩ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiuneq12df 5024 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
β„²π‘₯πœ‘    &   β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   (πœ‘ β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 = βˆͺ π‘₯ ∈ 𝐡 𝐷)
 
Theoremiuneq1d 5025* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 = βˆͺ π‘₯ ∈ 𝐡 𝐢)
 
Theoremiuneq12d 5026* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)
(πœ‘ β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 = βˆͺ π‘₯ ∈ 𝐡 𝐷)
 
Theoremiuneq2d 5027* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)
(πœ‘ β†’ 𝐡 = 𝐢)    β‡’   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremnfiun 5028* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2372. See nfiung 5030 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐴    &   β„²π‘¦π΅    β‡’   β„²π‘¦βˆͺ π‘₯ ∈ 𝐴 𝐡
 
Theoremnfiin 5029* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2372. See nfiing 5031 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐴    &   β„²π‘¦π΅    β‡’   β„²π‘¦βˆ© π‘₯ ∈ 𝐴 𝐡
 
Theoremnfiung 5030 Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfiun 5028 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
Ⅎ𝑦𝐴    &   β„²π‘¦π΅    β‡’   β„²π‘¦βˆͺ π‘₯ ∈ 𝐴 𝐡
 
Theoremnfiing 5031 Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2372. See nfiin 5029 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
Ⅎ𝑦𝐴    &   β„²π‘¦π΅    β‡’   β„²π‘¦βˆ© π‘₯ ∈ 𝐴 𝐡
 
Theoremnfiu1 5032 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)
β„²π‘₯βˆͺ π‘₯ ∈ 𝐴 𝐡
 
Theoremnfii1 5033 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
β„²π‘₯∩ π‘₯ ∈ 𝐴 𝐡
 
Theoremdfiun2g 5034* Alternate definition of indexed union when 𝐡 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.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 11-Dec-2024.)
(βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐢 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡})
 
Theoremdfiun2gOLD 5035* Obsolete version of dfiun2g 5034 as of 11-Dec-2024. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
(βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐢 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡})
 
Theoremdfiin2g 5036* Alternate definition of indexed intersection when 𝐡 is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
(βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐢 β†’ ∩ π‘₯ ∈ 𝐴 𝐡 = ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡})
 
Theoremdfiun2 5037* Alternate definition of indexed union when 𝐡 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
𝐡 ∈ V    β‡’   βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡}
 
Theoremdfiin2 5038* Alternate definition of indexed intersection when 𝐡 is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐡 ∈ V    β‡’   βˆ© π‘₯ ∈ 𝐴 𝐡 = ∩ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡}
 
Theoremdfiunv2 5039* Define double indexed union. (Contributed by FL, 6-Nov-2013.)
βˆͺ π‘₯ ∈ 𝐴 βˆͺ 𝑦 ∈ 𝐡 𝐢 = {𝑧 ∣ βˆƒπ‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 𝑧 ∈ 𝐢}
 
Theoremcbviun 5040* 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 2372. See cbviung 5042 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐡    &   β„²π‘₯𝐢    &   (π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviin 5041* 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 2372. See cbviing 5043 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐡    &   β„²π‘₯𝐢    &   (π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆ© π‘₯ ∈ 𝐴 𝐡 = ∩ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviung 5042* 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 2372. See cbviun 5040 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)
Ⅎ𝑦𝐡    &   β„²π‘₯𝐢    &   (π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviing 5043* Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2372. See cbviin 5041 for a version with more disjoint variable conditions, but not requiring ax-13 2372. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
Ⅎ𝑦𝐡    &   β„²π‘₯𝐢    &   (π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆ© π‘₯ ∈ 𝐴 𝐡 = ∩ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviunv 5044* 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 2372. See cbviunvg 5046 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviinv 5045* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2372. See cbviinvg 5047 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆ© π‘₯ ∈ 𝐴 𝐡 = ∩ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviunvg 5046* 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 2372. Usage of the weaker cbviunv 5044 is preferred. (Contributed by NM, 15-Sep-2003.) (New usage is discouraged.)
(π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆͺ π‘₯ ∈ 𝐴 𝐡 = βˆͺ 𝑦 ∈ 𝐴 𝐢
 
Theoremcbviinvg 5047* Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2372. Usage of the weaker cbviinv 5045 is preferred. (Contributed by Jeff Hankins, 26-Aug-2009.) (New usage is discouraged.)
(π‘₯ = 𝑦 β†’ 𝐡 = 𝐢)    β‡’   βˆ© π‘₯ ∈ 𝐴 𝐡 = ∩ 𝑦 ∈ 𝐴 𝐢
 
Theoremiunssf 5048 Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
β„²π‘₯𝐢    β‡’   (βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢 ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢)
 
Theoremiunss 5049* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢 ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢)
 
Theoremssiun 5050* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(βˆƒπ‘₯ ∈ 𝐴 𝐢 βŠ† 𝐡 β†’ 𝐢 βŠ† βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
Theoremssiun2 5051 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(π‘₯ ∈ 𝐴 β†’ 𝐡 βŠ† βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
Theoremssiun2s 5052* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)
(π‘₯ = 𝐢 β†’ 𝐡 = 𝐷)    β‡’   (𝐢 ∈ 𝐴 β†’ 𝐷 βŠ† βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
Theoremiunss2 5053* A subclass condition on the members of two indexed classes 𝐢(π‘₯) and 𝐷(𝑦) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4946. (Contributed by NM, 9-Dec-2004.)
(βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ 𝐡 𝐢 βŠ† 𝐷 β†’ βˆͺ π‘₯ ∈ 𝐴 𝐢 βŠ† βˆͺ 𝑦 ∈ 𝐡 𝐷)
 
Theoremiunssd 5054* Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 βŠ† 𝐢)    β‡’   (πœ‘ β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢)
 
Theoremiunab 5055* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
βˆͺ π‘₯ ∈ 𝐴 {𝑦 ∣ πœ‘} = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 πœ‘}
 
Theoremiunrab 5056* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
βˆͺ π‘₯ ∈ 𝐴 {𝑦 ∈ 𝐡 ∣ πœ‘} = {𝑦 ∈ 𝐡 ∣ βˆƒπ‘₯ ∈ 𝐴 πœ‘}
 
Theoremiunxdif2 5057* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(π‘₯ = 𝑦 β†’ 𝐢 = 𝐷)    β‡’   (βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ (𝐴 βˆ– 𝐡)𝐢 βŠ† 𝐷 β†’ βˆͺ 𝑦 ∈ (𝐴 βˆ– 𝐡)𝐷 = βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremssiinf 5058 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
β„²π‘₯𝐢    β‡’   (𝐢 βŠ† ∩ π‘₯ ∈ 𝐴 𝐡 ↔ βˆ€π‘₯ ∈ 𝐴 𝐢 βŠ† 𝐡)
 
Theoremssiin 5059* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐢 βŠ† ∩ π‘₯ ∈ 𝐴 𝐡 ↔ βˆ€π‘₯ ∈ 𝐴 𝐢 βŠ† 𝐡)
 
Theoremiinss 5060* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(βˆƒπ‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢 β†’ ∩ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐢)
 
Theoremiinss2 5061 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(π‘₯ ∈ 𝐴 β†’ ∩ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝐡)
 
Theoremuniiun 5062* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
βˆͺ 𝐴 = βˆͺ π‘₯ ∈ 𝐴 π‘₯
 
Theoremintiin 5063* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
∩ 𝐴 = ∩ π‘₯ ∈ 𝐴 π‘₯
 
Theoremiunid 5064* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.)
βˆͺ π‘₯ ∈ 𝐴 {π‘₯} = 𝐴
 
TheoremiunidOLD 5065* Obsolete version of iunid 5064 as of 15-Jan-2025. (Contributed by NM, 6-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
βˆͺ π‘₯ ∈ 𝐴 {π‘₯} = 𝐴
 
Theoremiun0 5066 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
βˆͺ π‘₯ ∈ 𝐴 βˆ… = βˆ…
 
Theorem0iun 5067 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
βˆͺ π‘₯ ∈ βˆ… 𝐴 = βˆ…
 
Theorem0iin 5068 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
∩ π‘₯ ∈ βˆ… 𝐴 = V
 
Theoremviin 5069* Indexed intersection with a universal index class. When 𝐴 doesn't depend on π‘₯, this evaluates to 𝐴 by 19.3 2196 and abid2 2872. When 𝐴 = π‘₯, this evaluates to βˆ… by intiin 5063 and intv 5363. (Contributed by NM, 11-Sep-2008.)
∩ π‘₯ ∈ V 𝐴 = {𝑦 ∣ βˆ€π‘₯ 𝑦 ∈ 𝐴}
 
Theoremiunsn 5070* Indexed union of a singleton. Compare dfiun2 5037 and rnmpt 5955. (Contributed by Steven Nguyen, 7-Jun-2023.)
βˆͺ π‘₯ ∈ 𝐴 {𝐡} = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑦 = 𝐡}
 
Theoremiunn0 5071* There is a nonempty class in an indexed collection 𝐡(π‘₯) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(βˆƒπ‘₯ ∈ 𝐴 𝐡 β‰  βˆ… ↔ βˆͺ π‘₯ ∈ 𝐴 𝐡 β‰  βˆ…)
 
Theoremiinab 5072* Indexed intersection of a class abstraction. (Contributed by NM, 6-Dec-2011.)
∩ π‘₯ ∈ 𝐴 {𝑦 ∣ πœ‘} = {𝑦 ∣ βˆ€π‘₯ ∈ 𝐴 πœ‘}
 
Theoremiinrab 5073* Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 {𝑦 ∈ 𝐡 ∣ πœ‘} = {𝑦 ∈ 𝐡 ∣ βˆ€π‘₯ ∈ 𝐴 πœ‘})
 
Theoremiinrab2 5074* Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
(∩ π‘₯ ∈ 𝐴 {𝑦 ∈ 𝐡 ∣ πœ‘} ∩ 𝐡) = {𝑦 ∈ 𝐡 ∣ βˆ€π‘₯ ∈ 𝐴 πœ‘}
 
Theoremiunin2 5075* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5062 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
βˆͺ π‘₯ ∈ 𝐴 (𝐡 ∩ 𝐢) = (𝐡 ∩ βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiunin1 5076* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5062 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
βˆͺ π‘₯ ∈ 𝐴 (𝐢 ∩ 𝐡) = (βˆͺ π‘₯ ∈ 𝐴 𝐢 ∩ 𝐡)
 
Theoremiinun2 5077* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5063 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
∩ π‘₯ ∈ 𝐴 (𝐡 βˆͺ 𝐢) = (𝐡 βˆͺ ∩ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiundif2 5078* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 5063 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
βˆͺ π‘₯ ∈ 𝐴 (𝐡 βˆ– 𝐢) = (𝐡 βˆ– ∩ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiindif1 5079* Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 (𝐡 βˆ– 𝐢) = (∩ π‘₯ ∈ 𝐴 𝐡 βˆ– 𝐢))
 
Theorem2iunin 5080* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
βˆͺ π‘₯ ∈ 𝐴 βˆͺ 𝑦 ∈ 𝐡 (𝐢 ∩ 𝐷) = (βˆͺ π‘₯ ∈ 𝐴 𝐢 ∩ βˆͺ 𝑦 ∈ 𝐡 𝐷)
 
Theoremiindif2 5081* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 5062 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 (𝐡 βˆ– 𝐢) = (𝐡 βˆ– βˆͺ π‘₯ ∈ 𝐴 𝐢))
 
Theoremiinin2 5082* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5063 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 (𝐡 ∩ 𝐢) = (𝐡 ∩ ∩ π‘₯ ∈ 𝐴 𝐢))
 
Theoremiinin1 5083* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5063 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 β‰  βˆ… β†’ ∩ π‘₯ ∈ 𝐴 (𝐢 ∩ 𝐡) = (∩ π‘₯ ∈ 𝐴 𝐢 ∩ 𝐡))
 
Theoremiinvdif 5084* The indexed intersection of a complement. (Contributed by GΓ©rard Lang, 5-Aug-2018.)
∩ π‘₯ ∈ 𝐴 (V βˆ– 𝐡) = (V βˆ– βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
Theoremelriin 5085* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝐡 ∈ (𝐴 ∩ ∩ π‘₯ ∈ 𝑋 𝑆) ↔ (𝐡 ∈ 𝐴 ∧ βˆ€π‘₯ ∈ 𝑋 𝐡 ∈ 𝑆))
 
Theoremriin0 5086* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = βˆ… β†’ (𝐴 ∩ ∩ π‘₯ ∈ 𝑋 𝑆) = 𝐴)
 
Theoremriinn0 5087* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((βˆ€π‘₯ ∈ 𝑋 𝑆 βŠ† 𝐴 ∧ 𝑋 β‰  βˆ…) β†’ (𝐴 ∩ ∩ π‘₯ ∈ 𝑋 𝑆) = ∩ π‘₯ ∈ 𝑋 𝑆)
 
Theoremriinrab 5088* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴 ∩ ∩ π‘₯ ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ πœ‘}) = {𝑦 ∈ 𝐴 ∣ βˆ€π‘₯ ∈ 𝑋 πœ‘}
 
Theoremsymdif0 5089 Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 β–³ βˆ…) = 𝐴
 
Theoremsymdifv 5090 The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 β–³ V) = (V βˆ– 𝐴)
 
Theoremsymdifid 5091 The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
(𝐴 β–³ 𝐴) = βˆ…
 
Theoremiinxsng 5092* 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.)
(π‘₯ = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (𝐴 ∈ 𝑉 β†’ ∩ π‘₯ ∈ {𝐴}𝐡 = 𝐢)
 
Theoremiinxprg 5093* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(π‘₯ = 𝐴 β†’ 𝐢 = 𝐷)    &   (π‘₯ = 𝐡 β†’ 𝐢 = 𝐸)    β‡’   ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ∩ π‘₯ ∈ {𝐴, 𝐡}𝐢 = (𝐷 ∩ 𝐸))
 
Theoremiunxsng 5094* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(π‘₯ = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (𝐴 ∈ 𝑉 β†’ βˆͺ π‘₯ ∈ {𝐴}𝐡 = 𝐢)
 
Theoremiunxsn 5095* 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    &   (π‘₯ = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   βˆͺ π‘₯ ∈ {𝐴}𝐡 = 𝐢
 
Theoremiunxsngf 5096* 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 2372. (Revised by Gino Giotto, 19-May-2023.)
β„²π‘₯𝐢    &   (π‘₯ = 𝐴 β†’ 𝐡 = 𝐢)    β‡’   (𝐴 ∈ 𝑉 β†’ βˆͺ π‘₯ ∈ {𝐴}𝐡 = 𝐢)
 
Theoremiunun 5097 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
βˆͺ π‘₯ ∈ 𝐴 (𝐡 βˆͺ 𝐢) = (βˆͺ π‘₯ ∈ 𝐴 𝐡 βˆͺ βˆͺ π‘₯ ∈ 𝐴 𝐢)
 
Theoremiunxun 5098 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
βˆͺ π‘₯ ∈ (𝐴 βˆͺ 𝐡)𝐢 = (βˆͺ π‘₯ ∈ 𝐴 𝐢 βˆͺ βˆͺ π‘₯ ∈ 𝐡 𝐢)
 
Theoremiunxdif3 5099* An indexed union where some terms are the empty set. See iunxdif2 5057. (Contributed by Thierry Arnoux, 4-May-2020.)
β„²π‘₯𝐸    β‡’   (βˆ€π‘₯ ∈ 𝐸 𝐡 = βˆ… β†’ βˆͺ π‘₯ ∈ (𝐴 βˆ– 𝐸)𝐡 = βˆͺ π‘₯ ∈ 𝐴 𝐡)
 
Theoremiunxprg 5100* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
(π‘₯ = 𝐴 β†’ 𝐢 = 𝐷)    &   (π‘₯ = 𝐡 β†’ 𝐢 = 𝐸)    β‡’   ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ βˆͺ π‘₯ ∈ {𝐴, 𝐡}𝐢 = (𝐷 βˆͺ 𝐸))
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