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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 23050 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld 23051 | . . 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 688 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 ∪ cuni 4912 ‘cfv 6563 Topctop 22915 Clsdccld 23040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-top 22916 df-cld 23043 |
This theorem is referenced by: difopn 23058 iincld 23063 uncld 23065 iuncld 23069 clsval2 23074 opncldf1 23108 opncldf3 23110 restcld 23196 lecldbas 23243 cnclima 23292 nrmsep2 23380 nrmsep 23381 regsep2 23400 cmpcld 23426 dfconn2 23443 txcld 23627 ptcld 23637 kqcldsat 23757 regr1lem 23763 filconn 23907 cldsubg 24135 limcnlp 25928 dvrec 26008 dvexp3 26031 lhop1lem 26067 abelth 26500 logdmopn 26706 lgamucov 27096 onsucconni 36420 onint1 36432 pibt2 37400 mblfinlem3 37646 mblfinlem4 37647 ismblfin 37648 dvtanlem 37656 dvasin 37691 dvacos 37692 dvreasin 37693 dvreacos 37694 readvrec2 42370 fourierdlem62 46124 opncldeqv 48698 iscnrm3rlem5 48741 |
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