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Mirrors > Home > MPE Home > Th. List > isopn2 | Structured version Visualization version GIF version |
Description: A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
isopn2 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4059 | . . . 4 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋 | |
2 | iscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld2 21633 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽)) |
4 | 1, 3 | mpan2 690 | . . 3 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽)) |
5 | dfss4 4185 | . . . . 5 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆) | |
6 | 5 | biimpi 219 | . . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆) |
7 | 6 | eleq1d 2874 | . . 3 ⊢ (𝑆 ⊆ 𝑋 → ((𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽)) |
8 | 4, 7 | sylan9bb 513 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽)) |
9 | 8 | bicomd 226 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 Clsdccld 21621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-cld 21624 |
This theorem is referenced by: opncld 21638 iscncl 21874 1stckgen 22159 txkgen 22257 qtoprest 22322 qtopcmap 22324 stoweidlem28 42670 |
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