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Theorem chss 22580
Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)
Assertion
Ref Expression
chss  |-  ( H  e.  CH  ->  H  C_ 
~H )

Proof of Theorem chss
StepHypRef Expression
1 chsh 22575 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 shss 22560 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2syl 16 1  |-  ( H  e.  CH  ->  H  C_ 
~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    C_ wss 3263   ~Hchil 22270   SHcsh 22279   CHcch 22280
This theorem is referenced by:  chel  22581  pjhcl  22751  dfch2  22757  shlub  22764  chsscon2  22852  chscllem2  22988  pjvec  23046  pjocvec  23047  pjhf  23058  elpjrn  23541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-hilex 22350
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fv 5402  df-ov 6023  df-sh 22557  df-ch 22572
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