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Theorem cldopn 21638
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 21633 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 21634 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simplbda 502 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋𝑆) ∈ 𝐽)
51, 4mpancom 686 1 (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  cdif 3932  wss 3935   cuni 4837  cfv 6354  Topctop 21500  Clsdccld 21623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fn 6357  df-fv 6362  df-top 21501  df-cld 21626
This theorem is referenced by:  difopn  21641  iincld  21646  uncld  21648  iuncld  21652  clsval2  21657  opncldf1  21691  opncldf3  21693  restcld  21779  lecldbas  21826  cnclima  21875  nrmsep2  21963  nrmsep  21964  regsep2  21983  cmpcld  22009  dfconn2  22026  txcld  22210  ptcld  22220  kqcldsat  22340  regr1lem  22346  filconn  22490  cldsubg  22718  limcnlp  24475  dvrec  24551  dvexp3  24574  lhop1lem  24609  abelth  25028  logdmopn  25231  lgamucov  25614  onsucconni  33785  onint1  33797  pibt2  34697  mblfinlem3  34930  mblfinlem4  34931  ismblfin  34932  dvtanlem  34940  dvasin  34977  dvacos  34978  dvreasin  34979  dvreacos  34980  fourierdlem62  42452
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