![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 484 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | eltopss 22728 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ⊆ 𝑋) |
4 | 2 | isopn2 22855 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) |
5 | 3, 4 | syldan 590 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽))) |
6 | 1, 5 | mpbid 231 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∖ cdif 3945 ⊆ wss 3948 ∪ cuni 4908 ‘cfv 6543 Topctop 22714 Clsdccld 22839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-top 22715 df-cld 22842 |
This theorem is referenced by: iincld 22862 iuncld 22868 clsval2 22873 cmntrcld 22886 elcls 22896 opncldf1 22907 opncldf2 22908 restcld 22995 iscncl 23092 pnrmopn 23166 isnrm2 23181 isnrm3 23182 isreg2 23200 hauscmplem 23229 conndisj 23239 hausllycmp 23317 1stckgen 23377 txkgen 23475 qtoprest 23540 qtopcmap 23542 icopnfcld 24603 lebnumlem1 24806 bcth3 25178 sxbrsigalem3 33734 pconnconn 34685 cvmscld 34727 cldbnd 35674 mblfinlem3 36990 mblfinlem4 36991 opncldeqv 47695 seposep 47719 |
Copyright terms: Public domain | W3C validator |