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Theorem chss 26547
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 26542 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 shss 26527 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2syl 17 1  |-  ( H  e.  CH  ->  H  C_ 
~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842    C_ wss 3413   ~Hchil 26236   SHcsh 26245   CHcch 26246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-hilex 26316
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fv 5576  df-ov 6280  df-sh 26524  df-ch 26539
This theorem is referenced by:  chel  26548  pjhcl  26719  dfch2  26725  shlub  26732  chsscon2  26820  chscllem2  26956  pjvec  27014  pjocvec  27015  pjhf  27026  elpjrn  27508
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