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Theorem cldopn 21243
Description: The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldopn (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)

Proof of Theorem cldopn
StepHypRef Expression
1 cldrcl 21238 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 21239 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simplbda 495 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋𝑆) ∈ 𝐽)
51, 4mpancom 678 1 (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cdif 3789  wss 3792   cuni 4671  cfv 6135  Topctop 21105  Clsdccld 21228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-top 21106  df-cld 21231
This theorem is referenced by:  difopn  21246  iincld  21251  uncld  21253  iuncld  21257  clsval2  21262  opncldf1  21296  opncldf3  21298  restcld  21384  lecldbas  21431  cnclima  21480  nrmsep2  21568  nrmsep  21569  regsep2  21588  cmpcld  21614  dfconn2  21631  txcld  21815  ptcld  21825  kqcldsat  21945  regr1lem  21951  filconn  22095  cldsubg  22322  limcnlp  24079  dvrec  24155  dvexp3  24178  lhop1lem  24213  abelth  24632  logdmopn  24832  lgamucov  25216  onsucconni  33019  onint1  33031  mblfinlem3  34076  mblfinlem4  34077  ismblfin  34078  dvtanlem  34086  dvasin  34123  dvacos  34124  dvreasin  34125  dvreacos  34126  fourierdlem62  41316
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