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Theorem cldopn 22398
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 22393 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 22394 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simplbda 501 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋𝑆) ∈ 𝐽)
51, 4mpancom 687 1 (𝑆 ∈ (Clsd‘𝐽) → (𝑋𝑆) ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cdif 3912  wss 3915   cuni 4870  cfv 6501  Topctop 22258  Clsdccld 22383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509  df-top 22259  df-cld 22386
This theorem is referenced by:  difopn  22401  iincld  22406  uncld  22408  iuncld  22412  clsval2  22417  opncldf1  22451  opncldf3  22453  restcld  22539  lecldbas  22586  cnclima  22635  nrmsep2  22723  nrmsep  22724  regsep2  22743  cmpcld  22769  dfconn2  22786  txcld  22970  ptcld  22980  kqcldsat  23100  regr1lem  23106  filconn  23250  cldsubg  23478  limcnlp  25258  dvrec  25335  dvexp3  25358  lhop1lem  25393  abelth  25816  logdmopn  26020  lgamucov  26403  onsucconni  34938  onint1  34950  pibt2  35917  mblfinlem3  36146  mblfinlem4  36147  ismblfin  36148  dvtanlem  36156  dvasin  36191  dvacos  36192  dvreasin  36193  dvreacos  36194  fourierdlem62  44483  opncldeqv  47008  iscnrm3rlem5  47051
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