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Theorem chss 21825
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 21820 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 shss 21805 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2syl 15 1  |-  ( H  e.  CH  ->  H  C_ 
~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    C_ wss 3165   ~Hchil 21515   SHcsh 21524   CHcch 21525
This theorem is referenced by:  chel  21826  pjhcl  21996  dfch2  22002  shlub  22009  chsscon2  22097  chscllem2  22233  pjvec  22291  pjocvec  22292  pjhf  22303  elpjrn  22786
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-rex 2562  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-uni 3844  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-iota 5235  df-fv 5279  df-ov 5877  df-sh 21802  df-ch 21817
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