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Theorem isopn2 20884
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 3770 . . . 4 (𝑋𝑆) ⊆ 𝑋
2 iscld.1 . . . . 5 𝑋 = 𝐽
32iscld2 20880 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋𝑆)) ∈ 𝐽))
41, 3mpan2 707 . . 3 (𝐽 ∈ Top → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋𝑆)) ∈ 𝐽))
5 dfss4 3891 . . . . 5 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
65biimpi 206 . . . 4 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76eleq1d 2715 . . 3 (𝑆𝑋 → ((𝑋 ∖ (𝑋𝑆)) ∈ 𝐽𝑆𝐽))
84, 7sylan9bb 736 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))
98bicomd 213 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  cdif 3604  wss 3607   cuni 4468  cfv 5926  Topctop 20746  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-top 20747  df-cld 20871
This theorem is referenced by:  opncld  20885  iscncl  21121  1stckgen  21405  txkgen  21503  qtoprest  21568  qtopcmap  21570  stoweidlem28  40563
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