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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iineq2 4901 | Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iuneq2i 4902 | Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 | ||
Theorem | iineq2i 4903 | Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 | ||
Theorem | iineq2d 4904 | Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.) |
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iuneq2dv 4905* | Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iineq2dv 4906* | Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iuneq12df 4907 | Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | iuneq1d 4908* | Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | iuneq12d 4909* | Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) | ||
Theorem | iuneq2d 4910* | Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | nfiun 4911* | Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2379. See nfiung 4913 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiin 4912* | Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2379. See nfiing 4914 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiung 4913 | Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfiun 4911 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiing 4914 | Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfiin 4912 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfiu1 4915 | Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.) |
⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfii1 4916 | Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.) |
⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | dfiun2g 4917* | 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.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | ||
Theorem | dfiun2gOLD 4918* | Obsolete proof of dfiun2g 4917 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.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | ||
Theorem | dfiin2g 4919* | Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Jeff Hankins, 27-Aug-2009.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | ||
Theorem | dfiun2 4920* | 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 ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | ||
Theorem | dfiin2 4921* | 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 ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | ||
Theorem | dfiunv2 4922* | Define double indexed union. (Contributed by FL, 6-Nov-2013.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶} | ||
Theorem | cbviun 4923* | 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 2379. See cbviung 4925 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviin 4924* | 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 2379. See cbviing 4926 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviung 4925* | 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 2379. See cbviun 4923 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviing 4926* | Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2379. See cbviin 4924 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviunv 4927* | 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 2379. See cbviunvg 4929 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviinv 4928* | Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2379. See cbviinvg 4930 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviunvg 4929* | 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 2379. Usage of the weaker cbviunv 4927 is preferred. (Contributed by NM, 15-Sep-2003.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbviinvg 4930* | Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2379. Usage of the weaker cbviinv 4928 is preferred. (Contributed by Jeff Hankins, 26-Aug-2009.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | ||
Theorem | iunssf 4931 | Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | iunss 4932* | Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | ssiun 4933* | Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | ssiun2 4934 | Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | ssiun2s 4935* | Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunss2 4936* | 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 4833. (Contributed by NM, 9-Dec-2004.) |
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷) | ||
Theorem | iunssd 4937* | Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | iunab 4938* | The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iunrab 4939* | The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iunxdif2 4940* | Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.) |
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ssiinf 4941 | Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.) |
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) | ||
Theorem | ssiin 4942* | Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.) |
⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) | ||
Theorem | iinss 4943* | Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) | ||
Theorem | iinss2 4944 | An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) | ||
Theorem | uniiun 4945* | Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.) |
⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | ||
Theorem | intiin 4946* | Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.) |
⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | ||
Theorem | iunid 4947* | An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) |
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | ||
Theorem | iun0 4948 | An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ | ||
Theorem | 0iun 4949 | An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ | ||
Theorem | 0iin 4950 | An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.) |
⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V | ||
Theorem | viin 4951* | Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2200 and abid2 2932. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 4946 and intv 5229. (Contributed by NM, 11-Sep-2008.) |
⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} | ||
Theorem | iunn0 4952* | 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.) |
⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | iinab 4953* | Indexed intersection of a class abstraction. (Contributed by NM, 6-Dec-2011.) |
⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iinrab 4954* | Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}) | ||
Theorem | iinrab2 4955* | Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.) |
⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑} | ||
Theorem | iunin2 4956* | Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4945 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iunin1 4957* | Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4945 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) | ||
Theorem | iinun2 4958* | Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4946 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.) |
⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iundif2 4959* | Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4946 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iindif1 4960* | Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶)) | ||
Theorem | 2iunin 4961* | Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.) |
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) | ||
Theorem | iindif2 4962* | Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4945 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)) | ||
Theorem | iinin2 4963* | Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4946 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)) | ||
Theorem | iinin1 4964* | Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4946 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.) |
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)) | ||
Theorem | iinvdif 4965* | The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.) |
⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | elriin 4966* | Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ (𝐵 ∈ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) | ||
Theorem | riin0 4967* | Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴) | ||
Theorem | riinn0 4968* | Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆) | ||
Theorem | riinrab 4969* | Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝜑}) = {𝑦 ∈ 𝐴 ∣ ∀𝑥 ∈ 𝑋 𝜑} | ||
Theorem | symdif0 4970 | 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 | ||
Theorem | symdifv 4971 | The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.) |
⊢ (𝐴 △ V) = (V ∖ 𝐴) | ||
Theorem | symdifid 4972 | The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.) |
⊢ (𝐴 △ 𝐴) = ∅ | ||
Theorem | iinxsng 4973* | 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 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶) | ||
Theorem | iinxprg 4974* | Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.) |
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸)) | ||
Theorem | iunxsng 4975* | A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) | ||
Theorem | iunxsn 4976* | 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 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 | ||
Theorem | iunxsngf 4977* | 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 2379. (Revised by Gino Giotto, 19-May-2023.) |
⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶) | ||
Theorem | iunun 4978 | Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) |
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ ∪ 𝑥 ∈ 𝐴 𝐶) | ||
Theorem | iunxun 4979 | 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 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪ 𝑥 ∈ 𝐴 𝐶 ∪ ∪ 𝑥 ∈ 𝐵 𝐶) | ||
Theorem | iunxdif3 4980* | An indexed union where some terms are the empty set. See iunxdif2 4940. (Contributed by Thierry Arnoux, 4-May-2020.) |
⊢ Ⅎ𝑥𝐸 ⇒ ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | iunxprg 4981* | A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.) |
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) & ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸)) | ||
Theorem | iunxiun 4982* | Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.) |
⊢ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 | ||
Theorem | iinuni 4983* | 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) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) | ||
Theorem | iununi 4984* | 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) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥)) | ||
Theorem | sspwuni 4985 | Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) | ||
Theorem | pwssb 4986* | Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | ||
Theorem | elpwpw 4987 | 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 ∧ ∪ 𝐴 ⊆ 𝐵)) | ||
Theorem | pwpwab 4988* | 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 ∣ ∪ 𝑥 ⊆ 𝐴} | ||
Theorem | pwpwssunieq 4989* | 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 | ||
Theorem | elpwuni 4990 | Relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) | ||
Theorem | iinpw 4991* | 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 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 | ||
Theorem | iunpwss 4992* | 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.) |
⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴 | ||
Theorem | intss2 4993 | 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 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋)) | ||
Theorem | rintn0 4994 | Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.) |
⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) | ||
Syntax | wdisj 4995 | Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥 ∈ 𝐴, is a disjoint family. |
wff Disj 𝑥 ∈ 𝐴 𝐵 | ||
Definition | df-disj 4996* | A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.) |
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | ||
Theorem | dfdisj2 4997* | Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.) |
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | ||
Theorem | disjss2 4998 | If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) | ||
Theorem | disjeq2 4999 | Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) | ||
Theorem | disjeq2dv 5000* | Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) |
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