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Theorem cmclsopn 21671
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 21659 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
32difeq2d 4053 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))))
4 difss 4062 . . . . . . 7 (𝑋𝑆) ⊆ 𝑋
51ntropn 21658 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
64, 5mpan2 690 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
71eltopss 21516 . . . . . 6 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽) → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
86, 7mpdan 686 . . . . 5 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
9 dfss4 4188 . . . . 5 (((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
108, 9sylib 221 . . . 4 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
1110, 6eqeltrd 2893 . . 3 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
1211adantr 484 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
133, 12eqeltrd 2893 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cdif 3881  wss 3884   cuni 4803  cfv 6328  Topctop 21502  intcnt 21626  clsccl 21627
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-top 21503  df-cld 21628  df-ntr 21629  df-cls 21630
This theorem is referenced by:  elcls  21682
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