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Mirrors > Home > MPE Home > Th. List > opncld | Structured version Visualization version GIF version |
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 476 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | eltopss 20537 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋) |
4 | 2 | isopn2 20646 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) |
5 | 3, 4 | syldan 486 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) |
6 | 1, 5 | mpbid 221 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 Topctop 20517 Clsdccld 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-top 20521 df-cld 20633 |
This theorem is referenced by: iincld 20653 iuncld 20659 clsval2 20664 cmntrcld 20677 elcls 20687 opncldf1 20698 opncldf2 20699 restcld 20786 iscncl 20883 pnrmopn 20957 isnrm2 20972 isnrm3 20973 isreg2 20991 hauscmplem 21019 conndisj 21029 hausllycmp 21107 1stckgen 21167 txkgen 21265 qtoprest 21330 qtopcmap 21332 icopnfcld 22381 lebnumlem1 22568 bcth3 22936 sxbrsigalem3 29661 pconcon 30467 cvmscld 30509 cldbnd 31491 mblfinlem3 32618 mblfinlem4 32619 |
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