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Theorem cmclsopn 22397
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 22385 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
32difeq2d 4080 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))))
4 difss 4089 . . . . . . 7 (𝑋𝑆) ⊆ 𝑋
51ntropn 22384 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
64, 5mpan2 689 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
71eltopss 22240 . . . . . 6 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽) → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
86, 7mpdan 685 . . . . 5 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
9 dfss4 4216 . . . . 5 (((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
108, 9sylib 217 . . . 4 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
1110, 6eqeltrd 2838 . . 3 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
1211adantr 481 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
133, 12eqeltrd 2838 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cdif 3905  wss 3908   cuni 4863  cfv 6493  Topctop 22226  intcnt 22352  clsccl 22353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22227  df-cld 22354  df-ntr 22355  df-cls 22356
This theorem is referenced by:  elcls  22408
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