| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 23034 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
| 2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
| 3 | 2 | iscld 23035 | . . 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 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 Clsdccld 23024 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-top 22900 df-cld 23027 |
| This theorem is referenced by: difopn 23042 iincld 23047 uncld 23049 iuncld 23053 clsval2 23058 opncldf1 23092 opncldf3 23094 restcld 23180 lecldbas 23227 cnclima 23276 nrmsep2 23364 nrmsep 23365 regsep2 23384 cmpcld 23410 dfconn2 23427 txcld 23611 ptcld 23621 kqcldsat 23741 regr1lem 23747 filconn 23891 cldsubg 24119 cnn0opn 24808 limcnlp 25913 lhop1lem 26052 abelth 26485 logdmopn 26691 lgamucov 27081 onsucconni 36438 onint1 36450 pibt2 37418 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 dvasin 37711 dvacos 37712 dvreasin 37713 dvreacos 37714 readvrec2 42391 fourierdlem62 46183 opncldeqv 48799 iscnrm3rlem5 48841 |
| Copyright terms: Public domain | W3C validator |