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Theorem cldopn 23055
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 23050 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 23051 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simplbda 499 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋𝑆) ∈ 𝐽)
51, 4mpancom 688 1 (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cdif 3960  wss 3963   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-top 22916  df-cld 23043
This theorem is referenced by:  difopn  23058  iincld  23063  uncld  23065  iuncld  23069  clsval2  23074  opncldf1  23108  opncldf3  23110  restcld  23196  lecldbas  23243  cnclima  23292  nrmsep2  23380  nrmsep  23381  regsep2  23400  cmpcld  23426  dfconn2  23443  txcld  23627  ptcld  23637  kqcldsat  23757  regr1lem  23763  filconn  23907  cldsubg  24135  limcnlp  25928  dvrec  26008  dvexp3  26031  lhop1lem  26067  abelth  26500  logdmopn  26706  lgamucov  27096  onsucconni  36420  onint1  36432  pibt2  37400  mblfinlem3  37646  mblfinlem4  37647  ismblfin  37648  dvtanlem  37656  dvasin  37691  dvacos  37692  dvreasin  37693  dvreacos  37694  readvrec2  42370  fourierdlem62  46124  opncldeqv  48698  iscnrm3rlem5  48741
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