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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem4 | Structured version Visualization version GIF version |
Description: Lemma for kur14 32465. 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 4237 | . 2 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴) | |
3 | 1, 2 | mpbi 232 | 1 ⊢ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 ∪ cuni 4840 ‘cfv 6357 Topctop 21503 intcnt 21627 clsccl 21628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 |
This theorem is referenced by: kur14lem7 32461 |
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