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Theorem cldss 22378
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 22375 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 22376 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simprbda 499 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
51, 4mpancom 686 1 (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cdif 3907  wss 3910   cuni 4865  cfv 6496  Topctop 22240  Clsdccld 22365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504  df-top 22241  df-cld 22368
This theorem is referenced by:  cldss2  22379  iincld  22388  uncld  22390  cldcls  22391  iuncld  22394  clsval2  22399  clsss3  22408  clsss2  22421  opncldf1  22433  restcldr  22523  lmcld  22652  nrmsep2  22705  nrmsep  22706  isnrm2  22707  regsep2  22725  cmpcld  22751  dfconn2  22768  conncompclo  22784  cldllycmp  22844  txcld  22952  ptcld  22962  imasncld  23040  kqcldsat  23082  kqnrmlem1  23092  kqnrmlem2  23093  nrmhmph  23143  ufildr  23280  metnrmlem1a  24219  metnrmlem1  24220  metnrmlem2  24221  metnrmlem3  24222  cnheiborlem  24315  cmetss  24678  bcthlem5  24690  cldssbrsiga  32777  clsun  34791  cldregopn  34794  pibt2  35879  mblfinlem3  36108  mblfinlem4  36109  ismblfin  36110  cmpfiiin  40998  kelac1  41368  stoweidlem18  44231  stoweidlem57  44270  restcls2lem  46917
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