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Mirrors > Home > MPE Home > Th. List > cldopn | Structured version Visualization version GIF version |
Description: The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cldopn | ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldrcl 21562 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld 21563 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
4 | 3 | simplbda 500 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
5 | 1, 4 | mpancom 684 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ⊆ wss 3933 ∪ cuni 4830 ‘cfv 6348 Topctop 21429 Clsdccld 21552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 df-top 21430 df-cld 21555 |
This theorem is referenced by: difopn 21570 iincld 21575 uncld 21577 iuncld 21581 clsval2 21586 opncldf1 21620 opncldf3 21622 restcld 21708 lecldbas 21755 cnclima 21804 nrmsep2 21892 nrmsep 21893 regsep2 21912 cmpcld 21938 dfconn2 21955 txcld 22139 ptcld 22149 kqcldsat 22269 regr1lem 22275 filconn 22419 cldsubg 22646 limcnlp 24403 dvrec 24479 dvexp3 24502 lhop1lem 24537 abelth 24956 logdmopn 25159 lgamucov 25542 onsucconni 33682 onint1 33694 pibt2 34580 mblfinlem3 34812 mblfinlem4 34813 ismblfin 34814 dvtanlem 34822 dvasin 34859 dvacos 34860 dvreasin 34861 dvreacos 34862 fourierdlem62 42330 |
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