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Theorem cmclsopn 20860
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 20848 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
32difeq2d 3726 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))))
4 difss 3735 . . . . . . 7 (𝑋𝑆) ⊆ 𝑋
51ntropn 20847 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
64, 5mpan2 707 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
71eltopss 20706 . . . . . 6 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽) → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
86, 7mpdan 702 . . . . 5 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
9 dfss4 3856 . . . . 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 2700 . . 3 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
1211adantr 481 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
133, 12eqeltrd 2700 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  cdif 3569  wss 3572   cuni 4434  cfv 5886  Topctop 20692  intcnt 20815  clsccl 20816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-top 20693  df-cld 20817  df-ntr 20818  df-cls 20819
This theorem is referenced by:  elcls  20871
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