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Theorem chssii 25813
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 25809 . 2  |-  H  e.  SH
32shssii 25794 1  |-  H  C_  ~H
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1762    C_ wss 3471   ~Hchil 25500   CHcch 25510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-hilex 25580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fv 5589  df-ov 6280  df-sh 25788  df-ch 25803
This theorem is referenced by:  cheli  25814  chelii  25815  hhsscms  25859  chocvali  25881  chm1i  26038  chsscon3i  26043  chsscon2i  26045  chjoi  26070  chj1i  26071  shjshsi  26074  sshhococi  26128  h1dei  26132  spansnpji  26160  spanunsni  26161  h1datomi  26163  spansnji  26228  pjfi  26286  riesz3i  26645  hmopidmpji  26735  pjoccoi  26761  pjinvari  26774  stcltr2i  26858  mdsymi  26994  mdcompli  27012  dmdcompli  27013
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