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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 22852 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld 22853 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
4 | 3 | simplbda 499 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
5 | 1, 4 | mpancom 685 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3937 ⊆ wss 3940 ∪ cuni 4899 ‘cfv 6533 Topctop 22717 Clsdccld 22842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fn 6536 df-fv 6541 df-top 22718 df-cld 22845 |
This theorem is referenced by: difopn 22860 iincld 22865 uncld 22867 iuncld 22871 clsval2 22876 opncldf1 22910 opncldf3 22912 restcld 22998 lecldbas 23045 cnclima 23094 nrmsep2 23182 nrmsep 23183 regsep2 23202 cmpcld 23228 dfconn2 23245 txcld 23429 ptcld 23439 kqcldsat 23559 regr1lem 23565 filconn 23709 cldsubg 23937 limcnlp 25729 dvrec 25809 dvexp3 25832 lhop1lem 25868 abelth 26295 logdmopn 26499 lgamucov 26886 onsucconni 35812 onint1 35824 pibt2 36788 mblfinlem3 37017 mblfinlem4 37018 ismblfin 37019 dvtanlem 37027 dvasin 37062 dvacos 37063 dvreasin 37064 dvreacos 37065 fourierdlem62 45369 opncldeqv 47722 iscnrm3rlem5 47765 |
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