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Theorem cldss 23053
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 23050 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 23051 . . 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 1537  wcel 2106  cdif 3960  wss 3963   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-top 22916  df-cld 23043
This theorem is referenced by:  cldss2  23054  iincld  23063  uncld  23065  cldcls  23066  iuncld  23069  clsval2  23074  clsss3  23083  clsss2  23096  opncldf1  23108  restcldr  23198  lmcld  23327  nrmsep2  23380  nrmsep  23381  isnrm2  23382  regsep2  23400  cmpcld  23426  dfconn2  23443  conncompclo  23459  cldllycmp  23519  txcld  23627  ptcld  23637  imasncld  23715  kqcldsat  23757  kqnrmlem1  23767  kqnrmlem2  23768  nrmhmph  23818  ufildr  23955  metnrmlem1a  24894  metnrmlem1  24895  metnrmlem2  24896  metnrmlem3  24897  cnheiborlem  25000  cmetss  25364  bcthlem5  25376  cldssbrsiga  34168  clsun  36311  cldregopn  36314  pibt2  37400  mblfinlem3  37646  mblfinlem4  37647  ismblfin  37648  cmpfiiin  42685  kelac1  43052  stoweidlem18  45974  stoweidlem57  46013  restcls2lem  48709
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