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Theorem isopn2 21244
Description: A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
isopn2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))

Proof of Theorem isopn2
StepHypRef Expression
1 difss 3959 . . . 4 (𝑋𝑆) ⊆ 𝑋
2 iscld.1 . . . . 5 𝑋 = 𝐽
32iscld2 21240 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋𝑆)) ∈ 𝐽))
41, 3mpan2 681 . . 3 (𝐽 ∈ Top → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋𝑆)) ∈ 𝐽))
5 dfss4 4084 . . . . 5 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
65biimpi 208 . . . 4 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76eleq1d 2843 . . 3 (𝑆𝑋 → ((𝑋 ∖ (𝑋𝑆)) ∈ 𝐽𝑆𝐽))
84, 7sylan9bb 505 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))
98bicomd 215 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  cdif 3788  wss 3791   cuni 4671  cfv 6135  Topctop 21105  Clsdccld 21228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-top 21106  df-cld 21231
This theorem is referenced by:  opncld  21245  iscncl  21481  1stckgen  21766  txkgen  21864  qtoprest  21929  qtopcmap  21931  stoweidlem28  41165
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