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Theorem cldopn 21567
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 21562 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 21563 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simplbda 500 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋𝑆) ∈ 𝐽)
51, 4mpancom 684 1 (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cdif 3930  wss 3933   cuni 4830  cfv 6348  Topctop 21429  Clsdccld 21552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-top 21430  df-cld 21555
This theorem is referenced by:  difopn  21570  iincld  21575  uncld  21577  iuncld  21581  clsval2  21586  opncldf1  21620  opncldf3  21622  restcld  21708  lecldbas  21755  cnclima  21804  nrmsep2  21892  nrmsep  21893  regsep2  21912  cmpcld  21938  dfconn2  21955  txcld  22139  ptcld  22149  kqcldsat  22269  regr1lem  22275  filconn  22419  cldsubg  22646  limcnlp  24403  dvrec  24479  dvexp3  24502  lhop1lem  24537  abelth  24956  logdmopn  25159  lgamucov  25542  onsucconni  33682  onint1  33694  pibt2  34580  mblfinlem3  34812  mblfinlem4  34813  ismblfin  34814  dvtanlem  34822  dvasin  34859  dvacos  34860  dvreasin  34861  dvreacos  34862  fourierdlem62  42330
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