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Theorem chss 24769
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 24764 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 shss 24749 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2syl 16 1  |-  ( H  e.  CH  ->  H  C_ 
~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    C_ wss 3428   ~Hchil 24458   SHcsh 24467   CHcch 24468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-hilex 24538
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-xp 4946  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fv 5526  df-ov 6195  df-sh 24746  df-ch 24761
This theorem is referenced by:  chel  24770  pjhcl  24941  dfch2  24947  shlub  24954  chsscon2  25042  chscllem2  25178  pjvec  25236  pjocvec  25237  pjhf  25248  elpjrn  25731
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