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Theorem opncld 21633
Description: The complement of an open set is closed. (Contributed by NM, 6-Oct-2006.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
opncld ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))

Proof of Theorem opncld
StepHypRef Expression
1 simpr 487 . 2 ((𝐽 ∈ Top ∧ 𝑆𝐽) → 𝑆𝐽)
2 iscld.1 . . . 4 𝑋 = 𝐽
32eltopss 21507 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝐽) → 𝑆𝑋)
42isopn2 21632 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
53, 4syldan 593 . 2 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
61, 5mpbid 234 1 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  cdif 3931  wss 3934   cuni 4830  cfv 6348  Topctop 21493  Clsdccld 21616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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-fv 6356  df-top 21494  df-cld 21619
This theorem is referenced by:  iincld  21639  iuncld  21645  clsval2  21650  cmntrcld  21663  elcls  21673  opncldf1  21684  opncldf2  21685  restcld  21772  iscncl  21869  pnrmopn  21943  isnrm2  21958  isnrm3  21959  isreg2  21977  hauscmplem  22006  conndisj  22016  hausllycmp  22094  1stckgen  22154  txkgen  22252  qtoprest  22317  qtopcmap  22319  icopnfcld  23368  lebnumlem1  23557  bcth3  23926  sxbrsigalem3  31518  pconnconn  32466  cvmscld  32508  cldbnd  33662  mblfinlem3  34918  mblfinlem4  34919
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