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Theorem cldss 21630
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 21627 . 2 (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2 iscld.1 . . . 4 𝑋 = 𝐽
32iscld 21628 . . 3 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
43simprbda 502 . 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 1538  wcel 2115  cdif 3916  wss 3919   cuni 4824  cfv 6343  Topctop 21494  Clsdccld 21617
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-top 21495  df-cld 21620
This theorem is referenced by:  cldss2  21631  iincld  21640  uncld  21642  cldcls  21643  iuncld  21646  clsval2  21651  clsss3  21660  clsss2  21673  opncldf1  21685  restcldr  21775  lmcld  21904  nrmsep2  21957  nrmsep  21958  isnrm2  21959  regsep2  21977  cmpcld  22003  dfconn2  22020  conncompclo  22036  cldllycmp  22096  txcld  22204  ptcld  22214  imasncld  22292  kqcldsat  22334  kqnrmlem1  22344  kqnrmlem2  22345  nrmhmph  22395  ufildr  22532  metnrmlem1a  23459  metnrmlem1  23460  metnrmlem2  23461  metnrmlem3  23462  cnheiborlem  23555  cmetss  23916  bcthlem5  23928  cldssbrsiga  31471  clsun  33701  cldregopn  33704  pibt2  34746  mblfinlem3  35006  mblfinlem4  35007  ismblfin  35008  cmpfiiin  39491  kelac1  39860  stoweidlem18  42523  stoweidlem57  42562
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