MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opncld Structured version   Visualization version   GIF version

Theorem opncld 21163
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 478 . 2 ((𝐽 ∈ Top ∧ 𝑆𝐽) → 𝑆𝐽)
2 iscld.1 . . . 4 𝑋 = 𝐽
32eltopss 21037 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝐽) → 𝑆𝑋)
42isopn2 21162 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
53, 4syldan 586 . 2 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
61, 5mpbid 224 1 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑋𝑆) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cdif 3764  wss 3767   cuni 4626  cfv 6099  Topctop 21023  Clsdccld 21146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-top 21024  df-cld 21149
This theorem is referenced by:  iincld  21169  iuncld  21175  clsval2  21180  cmntrcld  21193  elcls  21203  opncldf1  21214  opncldf2  21215  restcld  21302  iscncl  21399  pnrmopn  21473  isnrm2  21488  isnrm3  21489  isreg2  21507  hauscmplem  21535  conndisj  21545  hausllycmp  21623  1stckgen  21683  txkgen  21781  qtoprest  21846  qtopcmap  21848  icopnfcld  22896  lebnumlem1  23085  bcth3  23454  sxbrsigalem3  30842  pconnconn  31722  cvmscld  31764  cldbnd  32825  mblfinlem3  33929  mblfinlem4  33930
  Copyright terms: Public domain W3C validator