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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 21238 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld 21239 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
4 | 3 | simplbda 495 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
5 | 1, 4 | mpancom 678 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ⊆ wss 3792 ∪ cuni 4671 ‘cfv 6135 Topctop 21105 Clsdccld 21228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fn 6138 df-fv 6143 df-top 21106 df-cld 21231 |
This theorem is referenced by: difopn 21246 iincld 21251 uncld 21253 iuncld 21257 clsval2 21262 opncldf1 21296 opncldf3 21298 restcld 21384 lecldbas 21431 cnclima 21480 nrmsep2 21568 nrmsep 21569 regsep2 21588 cmpcld 21614 dfconn2 21631 txcld 21815 ptcld 21825 kqcldsat 21945 regr1lem 21951 filconn 22095 cldsubg 22322 limcnlp 24079 dvrec 24155 dvexp3 24178 lhop1lem 24213 abelth 24632 logdmopn 24832 lgamucov 25216 onsucconni 33019 onint1 33031 mblfinlem3 34076 mblfinlem4 34077 ismblfin 34078 dvtanlem 34086 dvasin 34123 dvacos 34124 dvreasin 34125 dvreacos 34126 fourierdlem62 41316 |
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