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Theorem shssii 25903
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
shssi.1  |-  H  e.  SH
Assertion
Ref Expression
shssii  |-  H  C_  ~H

Proof of Theorem shssii
StepHypRef Expression
1 shssi.1 . 2  |-  H  e.  SH
2 shss 25900 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2ax-mp 5 1  |-  H  C_  ~H
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767    C_ wss 3476   ~Hchil 25609   SHcsh 25618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-sh 25897
This theorem is referenced by:  sheli  25904  shelii  25905  chssii  25922  hhssabloi  25951  hhssnv  25953  hhssba  25960  shunssji  26060  shsval3i  26079  shjshsi  26183  span0  26233  spanuni  26235  imaelshi  26750  nlelchi  26753  hmopidmchi  26843  pjimai  26868  shatomistici  27053
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