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Theorem chssii 21811
Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chssi.1  |-  H  e. 
CH
Assertion
Ref Expression
chssii  |-  H  C_  ~H

Proof of Theorem chssii
StepHypRef Expression
1 chssi.1 . . 3  |-  H  e. 
CH
21chshii 21807 . 2  |-  H  e.  SH
32shssii 21792 1  |-  H  C_  ~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   ~Hchil 21499   CHcch 21509
This theorem is referenced by:  cheli  21812  chelii  21813  hhsscms  21856  chocvali  21878  chm1i  22035  chsscon3i  22040  chsscon2i  22042  chjoi  22067  chj1i  22068  shjshsi  22071  sshhococi  22125  h1dei  22129  spansnpji  22157  spanunsni  22158  h1datomi  22160  spansnji  22225  pjfi  22283  riesz3i  22642  hmopidmpji  22732  pjoccoi  22758  pjinvari  22771  stcltr2i  22855  mdsymi  22991  mdcompli  23009  dmdcompli  23010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-sh 21786  df-ch 21801
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