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Theorem cmclsopn 20914
Description: The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cmclsopn ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)

Proof of Theorem cmclsopn
StepHypRef Expression
1 clscld.1 . . . 4 𝑋 = 𝐽
21clsval2 20902 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
32difeq2d 3761 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))))
4 difss 3770 . . . . . . 7 (𝑋𝑆) ⊆ 𝑋
51ntropn 20901 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
64, 5mpan2 707 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
71eltopss 20760 . . . . . 6 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽) → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
86, 7mpdan 703 . . . . 5 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
9 dfss4 3891 . . . . 5 (((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
108, 9sylib 208 . . . 4 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
1110, 6eqeltrd 2730 . . 3 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
1211adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
133, 12eqeltrd 2730 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cdif 3604  wss 3607   cuni 4468  cfv 5926  Topctop 20746  intcnt 20869  clsccl 20870
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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
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-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-int 4508  df-iun 4554  df-iin 4555  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-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by:  elcls  20925
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