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Theorem cldss 23037
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 23034 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 23035 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simprbda 498 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
51, 4mpancom 688 1 (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cdif 3948  wss 3951   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-top 22900  df-cld 23027
This theorem is referenced by:  cldss2  23038  iincld  23047  uncld  23049  cldcls  23050  iuncld  23053  clsval2  23058  clsss3  23067  clsss2  23080  opncldf1  23092  restcldr  23182  lmcld  23311  nrmsep2  23364  nrmsep  23365  isnrm2  23366  regsep2  23384  cmpcld  23410  dfconn2  23427  conncompclo  23443  cldllycmp  23503  txcld  23611  ptcld  23621  imasncld  23699  kqcldsat  23741  kqnrmlem1  23751  kqnrmlem2  23752  nrmhmph  23802  ufildr  23939  metnrmlem1a  24880  metnrmlem1  24881  metnrmlem2  24882  metnrmlem3  24883  cnheiborlem  24986  cmetss  25350  bcthlem5  25362  cldssbrsiga  34188  clsun  36329  cldregopn  36332  pibt2  37418  mblfinlem3  37666  mblfinlem4  37667  ismblfin  37668  cmpfiiin  42708  kelac1  43075  stoweidlem18  46033  stoweidlem57  46072  restcls2lem  48810
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