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Theorem cldss 20881
 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 20878 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 20879 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simprbda 652 . 2 ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆𝑋)
51, 4mpancom 704 1 (𝑆 ∈ (Clsd‘𝐽) → 𝑆𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ∖ cdif 3604   ⊆ wss 3607  ∪ cuni 4468  ‘cfv 5926  Topctop 20746  Clsdccld 20868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-top 20747  df-cld 20871 This theorem is referenced by:  cldss2  20882  iincld  20891  uncld  20893  cldcls  20894  iuncld  20897  clsval2  20902  clsss3  20911  clsss2  20924  opncldf1  20936  restcldr  21026  lmcld  21155  nrmsep2  21208  nrmsep  21209  isnrm2  21210  regsep2  21228  cmpcld  21253  dfconn2  21270  conncompclo  21286  cldllycmp  21346  txcld  21454  ptcld  21464  imasncld  21542  kqcldsat  21584  kqnrmlem1  21594  kqnrmlem2  21595  nrmhmph  21645  ufildr  21782  metnrmlem1a  22708  metnrmlem1  22709  metnrmlem2  22710  metnrmlem3  22711  cnheiborlem  22800  cmetss  23159  bcthlem5  23171  cldssbrsiga  30378  clsun  32448  cldregopn  32451  mblfinlem3  33578  mblfinlem4  33579  ismblfin  33580  cmpfiiin  37577  kelac1  37950  stoweidlem18  40553  stoweidlem57  40592
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