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Theorem chss 25809
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 25804 . 2  |-  ( H  e.  CH  ->  H  e.  SH )
2 shss 25789 . 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 1762    C_ wss 3469   ~Hchil 25498   SHcsh 25507   CHcch 25508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-hilex 25578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fv 5587  df-ov 6278  df-sh 25786  df-ch 25801
This theorem is referenced by:  chel  25810  pjhcl  25981  dfch2  25987  shlub  25994  chsscon2  26082  chscllem2  26218  pjvec  26276  pjocvec  26277  pjhf  26288  elpjrn  26771
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