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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem4 | Structured version Visualization version GIF version |
Description: Lemma for kur14 35187. 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 4288 | . 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | |
3 | 1, 2 | mpbi 230 | 1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 ∖ cdif 3973 ⊆ wss 3976 ∪ cuni 4937 ‘cfv 6579 Topctop 22924 intcnt 23050 clsccl 23051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3440 df-v 3486 df-dif 3979 df-in 3983 df-ss 3993 |
This theorem is referenced by: kur14lem7 35183 |
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