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Theorem shssii 21808
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 21805 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2ax-mp 8 1  |-  H  C_  ~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1696    C_ wss 3165   ~Hchil 21515   SHcsh 21524
This theorem is referenced by:  sheli  21809  shelii  21810  chssii  21827  hhssabloi  21855  hhssnv  21857  hhssba  21864  shunssji  21964  shsval3i  21983  shjshsi  22087  span0  22137  spanuni  22139  imaelshi  22654  nlelchi  22657  hmopidmchi  22747  pjimai  22772  shatomistici  22957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-sh 21802
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