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Theorem cldss 22308
Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldss (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)

Proof of Theorem cldss
StepHypRef Expression
1 cldrcl 22305 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 22306 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simprbda 500 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
51, 4mpancom 687 1 (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cdif 3906  wss 3909   cuni 4864  cfv 6492  Topctop 22170  Clsdccld 22295
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6444  df-fun 6494  df-fn 6495  df-fv 6500  df-top 22171  df-cld 22298
This theorem is referenced by:  cldss2  22309  iincld  22318  uncld  22320  cldcls  22321  iuncld  22324  clsval2  22329  clsss3  22338  clsss2  22351  opncldf1  22363  restcldr  22453  lmcld  22582  nrmsep2  22635  nrmsep  22636  isnrm2  22637  regsep2  22655  cmpcld  22681  dfconn2  22698  conncompclo  22714  cldllycmp  22774  txcld  22882  ptcld  22892  imasncld  22970  kqcldsat  23012  kqnrmlem1  23022  kqnrmlem2  23023  nrmhmph  23073  ufildr  23210  metnrmlem1a  24149  metnrmlem1  24150  metnrmlem2  24151  metnrmlem3  24152  cnheiborlem  24245  cmetss  24608  bcthlem5  24620  cldssbrsiga  32566  clsun  34731  cldregopn  34734  pibt2  35819  mblfinlem3  36048  mblfinlem4  36049  ismblfin  36050  cmpfiiin  40922  kelac1  41292  stoweidlem18  44050  stoweidlem57  44089  restcls2lem  46736
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