![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iscld2 | Structured version Visualization version GIF version |
Description: A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
iscld2 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑆) ∈ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | iscld 21202 | . 2 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
3 | 2 | baibd 535 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑆) ∈ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∖ cdif 3795 ⊆ wss 3798 ∪ cuni 4658 ‘cfv 6123 Topctop 21068 Clsdccld 21191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-top 21069 df-cld 21194 |
This theorem is referenced by: isopn2 21207 0cld 21213 uncld 21216 isclo 21262 cnclima 21443 ist1-2 21522 hausdiag 21819 qtopcld 21887 ufildr 22105 blcld 22680 icccld 22940 iocmnfcld 22942 zcld 22986 recld2 22987 qtophaus 30437 kelac2 38471 stoweidlem50 41054 |
Copyright terms: Public domain | W3C validator |