Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem4 | Structured version Visualization version GIF version |
Description: Lemma for kur14 32891. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem.j | ⊢ 𝐽 ∈ Top |
kur14lem.x | ⊢ 𝑋 = ∪ 𝐽 |
kur14lem.k | ⊢ 𝐾 = (cls‘𝐽) |
kur14lem.i | ⊢ 𝐼 = (int‘𝐽) |
kur14lem.a | ⊢ 𝐴 ⊆ 𝑋 |
Ref | Expression |
---|---|
kur14lem4 | ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem.a | . 2 ⊢ 𝐴 ⊆ 𝑋 | |
2 | dfss4 4173 | . 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | |
3 | 1, 2 | mpbi 233 | 1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 ⊆ wss 3866 ∪ cuni 4819 ‘cfv 6380 Topctop 21790 intcnt 21914 clsccl 21915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-in 3873 df-ss 3883 |
This theorem is referenced by: kur14lem7 32887 |
Copyright terms: Public domain | W3C validator |