Home | Metamath
Proof Explorer Theorem List (p. 49 of 465) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-29267) |
Hilbert Space Explorer
(29268-30790) |
Users' Mathboxes
(30791-46478) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elpr2elpr 4801* | For an element 𝐴 of an unordered pair which is a subset of a given set 𝑉, there is another (maybe the same) element 𝑏 of the given set 𝑉 being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}) | ||
Theorem | dfopif 4802 | Rewrite df-op 4570 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 1-Aug-2024.) (Avoid depending on this detail.) |
⊢ 〈𝐴, 𝐵〉 = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | ||
Theorem | dfopifOLD 4803 | Obsolete version of dfopif 4802 as of 1-Aug-2024. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 〈𝐴, 𝐵〉 = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | ||
Theorem | dfopg 4804 | Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | ||
Theorem | dfop 4805 | Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} | ||
Theorem | opeq1 4806 | Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | ||
Theorem | opeq2 4807 | Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | ||
Theorem | opeq12 4808 | Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → 〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉) | ||
Theorem | opeq1i 4809 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉 | ||
Theorem | opeq2i 4810 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉 | ||
Theorem | opeq12i 4811 | Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 | ||
Theorem | opeq1d 4812 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) | ||
Theorem | opeq2d 4813 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | ||
Theorem | opeq12d 4814 | Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | ||
Theorem | oteq1 4815 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) | ||
Theorem | oteq2 4816 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | ||
Theorem | oteq3 4817 | Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | ||
Theorem | oteq1d 4818 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐷〉 = 〈𝐵, 𝐶, 𝐷〉) | ||
Theorem | oteq2d 4819 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | ||
Theorem | oteq3d 4820 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 〈𝐶, 𝐷, 𝐴〉 = 〈𝐶, 𝐷, 𝐵〉) | ||
Theorem | oteq123d 4821 | Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) & ⊢ (𝜑 → 𝐸 = 𝐹) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) | ||
Theorem | nfop 4822 | Bound-variable hypothesis builder for ordered pairs. (Contributed by NM, 14-Nov-1995.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥〈𝐴, 𝐵〉 | ||
Theorem | nfopd 4823 | Deduction version of bound-variable hypothesis builder nfop 4822. This shows how the deduction version of a not-free theorem such as nfop 4822 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) & ⊢ (𝜑 → Ⅎ𝑥𝐵) ⇒ ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | ||
Theorem | csbopg 4824 | Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.) (Revised by ML, 25-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌〈𝐶, 𝐷〉 = 〈⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷〉) | ||
Theorem | opidg 4825 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Closed form of opid 4826. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.) |
⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) | ||
Theorem | opid 4826 | The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Inference form of opidg 4825. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} | ||
Theorem | ralunsn 4827* | Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) | ||
Theorem | 2ralunsn 4828* | Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))) | ||
Theorem | opprc 4829 | 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〉 = ∅) | ||
Theorem | opprc1 4830 | Expansion of an ordered pair when the first member is a proper class. See also opprc 4829. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (¬ 𝐴 ∈ V → 〈𝐴, 𝐵〉 = ∅) | ||
Theorem | opprc2 4831 | Expansion of an ordered pair when the second member is a proper class. See also opprc 4829. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (¬ 𝐵 ∈ V → 〈𝐴, 𝐵〉 = ∅) | ||
Theorem | oprcl 4832 | 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)) | ||
Theorem | pwsn 4833 | The power set of a singleton. (Contributed by NM, 5-Jun-2006.) |
⊢ 𝒫 {𝐴} = {∅, {𝐴}} | ||
Theorem | pwsnOLD 4834 | Obsolete version of pwsn 4833 as of 14-Apr-2024. Note that the proof is essentially the same once one inlines sssn 4761 in the proof of pwsn 4833. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝒫 {𝐴} = {∅, {𝐴}} | ||
Theorem | pwpr 4835 | The power set of an unordered pair. (Contributed by NM, 1-May-2009.) |
⊢ 𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) | ||
Theorem | pwtp 4836 | 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}})) | ||
Theorem | pwpwpw0 4837 | Compute the power set of the power set of the power set of the empty set. (See also pw0 4747 and pwpw0 4748.) (Contributed by NM, 2-May-2009.) |
⊢ 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) | ||
Theorem | pwv 4838 |
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 7224. (Contributed by NM, 14-Sep-2003.) |
⊢ 𝒫 V = V | ||
Theorem | prproe 4839* | 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} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}) | ||
Theorem | 3elpr2eq 4840 | 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}) ∧ (𝑌 ≠ 𝑋 ∧ 𝑍 ≠ 𝑋)) → 𝑌 = 𝑍) | ||
Syntax | cuni 4841 | Extend class notation to include the union of a class. Read: "union (of) 𝐴". |
class ∪ 𝐴 | ||
Definition | df-uni 4842* | 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 28777). This is similar to the union of two classes df-un 3893. (Contributed by NM, 23-Aug-1993.) |
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} | ||
Theorem | dfuni2 4843* | Alternate definition of class union. (Contributed by NM, 28-Jun-1998.) |
⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | ||
Theorem | eluni 4844* | Membership in class union. (Contributed by NM, 22-May-1994.) |
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) | ||
Theorem | eluni2 4845* | Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.) |
⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) | ||
Theorem | elunii 4846 | Membership in class union. (Contributed by NM, 24-Mar-1995.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ ∪ 𝐶) | ||
Theorem | nfunid 4847 | Deduction version of nfuni 4848. (Contributed by NM, 18-Feb-2013.) |
⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥∪ 𝐴) | ||
Theorem | nfuni 4848 | Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥∪ 𝐴 | ||
Theorem | uniss 4849 | 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.) |
⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵) | ||
Theorem | unissi 4850 | Subclass relationship for subclass union. Inference form of uniss 4849. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ∪ 𝐴 ⊆ ∪ 𝐵 | ||
Theorem | unissd 4851 | Subclass relationship for subclass union. Deduction form of uniss 4849. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 ⊆ ∪ 𝐵) | ||
Theorem | unieq 4852 | 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) | ||
Theorem | unieqOLD 4853 | Obsolete version of unieq 4852 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) | ||
Theorem | unieqi 4854 | Inference of equality of two class unions. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∪ 𝐴 = ∪ 𝐵 | ||
Theorem | unieqd 4855 | Deduction of equality of two class unions. (Contributed by NM, 21-Apr-1995.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝐴 = ∪ 𝐵) | ||
Theorem | eluniab 4856* | Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.) |
⊢ (𝐴 ∈ ∪ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | elunirab 4857* | Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.) |
⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝜑)) | ||
Theorem | uniprg 4858 | The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) Avoid using unipr 4859 to prove it from uniprg 4858. (Revised by BJ, 1-Sep-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | ||
Theorem | unipr 4859 | The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) (Proof shortened by BJ, 1-Sep-2024.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) | ||
Theorem | uniprOLD 4860 | Obsolete version of unipr 4859 as of 1-Sep-2024. (Contributed by NM, 23-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵) | ||
Theorem | uniprgOLD 4861 | Obsolete version of unipr 4859 as of 1-Sep-2024. (Contributed by NM, 25-Aug-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | ||
Theorem | unisng 4862 | A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.) |
⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴) | ||
Theorem | unisn 4863 | 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 | ||
Theorem | unisn3 4864* | Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.) |
⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴) | ||
Theorem | dfnfc2 4865* | 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 𝐴 ∈ 𝑉 → (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 = 𝐴)) | ||
Theorem | uniun 4866 | The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.) |
⊢ ∪ (𝐴 ∪ 𝐵) = (∪ 𝐴 ∪ ∪ 𝐵) | ||
Theorem | uniin 4867 | The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8575 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵) | ||
Theorem | ssuni 4868 | 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.) |
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ⊆ ∪ 𝐶) | ||
Theorem | uni0b 4869 | 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 ⊆ {∅}) | ||
Theorem | uni0c 4870* | The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.) |
⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅) | ||
Theorem | uni0 4871 | 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 5230. (Revised by Eric Schmidt, 4-Apr-2007.) |
⊢ ∪ ∅ = ∅ | ||
Theorem | csbuni 4872 | 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 / 𝑥⦌𝐵 | ||
Theorem | elssuni 4873 | 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.) |
⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | ||
Theorem | unissel 4874 | Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.) |
⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵) | ||
Theorem | unissb 4875* | Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) |
⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | ||
Theorem | uniss2 4876* | 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 4980 for a generalization to indexed unions. (Contributed by NM, 22-Mar-2004.) |
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ 𝐵) | ||
Theorem | unidif 4877* | If the difference 𝐴 ∖ 𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.) |
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴) | ||
Theorem | ssunieq 4878* | Relationship implying union. (Contributed by NM, 10-Nov-1999.) |
⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) | ||
Theorem | unimax 4879* | Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.) |
⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} = 𝐴) | ||
Theorem | pwuni 4880 | 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 | ||
Syntax | cint 4881 | Extend class notation to include the intersection of a class. Read: "intersection (of) 𝐴". |
class ∩ 𝐴 | ||
Definition | df-int 4882* | 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 3895. (Contributed by NM, 18-Aug-1993.) |
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} | ||
Theorem | dfint2 4883* | Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.) |
⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | ||
Theorem | inteq 4884 | Equality law for intersection. (Contributed by NM, 13-Sep-1999.) |
⊢ (𝐴 = 𝐵 → ∩ 𝐴 = ∩ 𝐵) | ||
Theorem | inteqi 4885 | Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ∩ 𝐴 = ∩ 𝐵 | ||
Theorem | inteqd 4886 | Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∩ 𝐴 = ∩ 𝐵) | ||
Theorem | elint 4887* | Membership in class intersection. (Contributed by NM, 21-May-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝐴 ∈ 𝑥)) | ||
Theorem | elint2 4888* | Membership in class intersection. (Contributed by NM, 14-Oct-1999.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥) | ||
Theorem | elintg 4889* | 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.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) | ||
Theorem | elinti 4890 | Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) | ||
Theorem | nfint 4891 | Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥∩ 𝐴 | ||
Theorem | elintab 4892* | Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)) | ||
Theorem | elintrab 4893* | Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥)) | ||
Theorem | elintrabg 4894* | Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) | ||
Theorem | int0 4895 | 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 | ||
Theorem | intss1 4896 | 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.) |
⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) | ||
Theorem | ssint 4897* | Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.) |
⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥) | ||
Theorem | ssintab 4898* | Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ (𝐴 ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ⊆ 𝑥)) | ||
Theorem | ssintub 4899* | Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.) |
⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} | ||
Theorem | ssmin 4900* | Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.) |
⊢ 𝐴 ⊆ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ 𝜑)} |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |