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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem4 | Structured version Visualization version GIF version |
Description: Lemma for kur14 34145. 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 4257 | . 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | |
3 | 1, 2 | mpbi 229 | 1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∖ cdif 3944 ⊆ wss 3947 ∪ cuni 4907 ‘cfv 6540 Topctop 22377 intcnt 22503 clsccl 22504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3950 df-in 3954 df-ss 3964 |
This theorem is referenced by: kur14lem7 34141 |
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