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Theorem chssii 26563
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 26559 . 2  |-  H  e.  SH
32shssii 26544 1  |-  H  C_  ~H
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1842    C_ wss 3414   ~Hchil 26250   CHcch 26260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-hilex 26330
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fv 5577  df-ov 6281  df-sh 26538  df-ch 26553
This theorem is referenced by:  cheli  26564  chelii  26565  hhsscms  26609  chocvali  26631  chm1i  26788  chsscon3i  26793  chsscon2i  26795  chjoi  26820  chj1i  26821  shjshsi  26824  sshhococi  26878  h1dei  26882  spansnpji  26910  spanunsni  26911  h1datomi  26913  spansnji  26978  pjfi  27036  riesz3i  27394  hmopidmpji  27484  pjoccoi  27510  pjinvari  27523  stcltr2i  27607  mdsymi  27743  mdcompli  27761  dmdcompli  27762
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