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| Description: The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.) | 
| Ref | Expression | 
|---|---|
| iscld.1 | ⊢ 𝑋 = ∪ 𝐽 | 
| Ref | Expression | 
|---|---|
| opncld | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpr 484 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
| 2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
| 3 | 2 | eltopss 22914 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋) | 
| 4 | 2 | isopn2 23041 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) | 
| 5 | 3, 4 | syldan 591 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) | 
| 6 | 1, 5 | mpbid 232 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 ⊆ wss 3950 ∪ cuni 4906 ‘cfv 6560 Topctop 22900 Clsdccld 23025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-top 22901 df-cld 23028 | 
| This theorem is referenced by: iincld 23048 iuncld 23054 clsval2 23059 cmntrcld 23072 elcls 23082 opncldf1 23093 opncldf2 23094 restcld 23181 iscncl 23278 pnrmopn 23352 isnrm2 23367 isnrm3 23368 isreg2 23386 hauscmplem 23415 conndisj 23425 hausllycmp 23503 1stckgen 23563 txkgen 23661 qtoprest 23726 qtopcmap 23728 icopnfcld 24789 lebnumlem1 24994 bcth3 25366 sxbrsigalem3 34275 pconnconn 35237 cvmscld 35279 cldbnd 36328 mblfinlem3 37667 mblfinlem4 37668 opncldeqv 48806 seposep 48830 | 
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