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Theorem sheli 25904
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
shssi.1  |-  H  e.  SH
Assertion
Ref Expression
sheli  |-  ( A  e.  H  ->  A  e.  ~H )

Proof of Theorem sheli
StepHypRef Expression
1 shssi.1 . . 3  |-  H  e.  SH
21shssii 25903 . 2  |-  H  C_  ~H
32sseli 3500 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ~Hchil 25609   SHcsh 25618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-sh 25897
This theorem is referenced by:  norm1exi  25941  hhssabloi  25951  hhssnv  25953  shscli  26008  shunssi  26059  shmodsi  26080  omlsii  26094  5oalem1  26345  5oalem2  26346  5oalem3  26347  5oalem5  26349  imaelshi  26750  pjimai  26868  shatomici  27050  shatomistici  27053  cdjreui  27124  cdj1i  27125  cdj3lem1  27126  cdj3lem2b  27129  cdj3lem3  27130  cdj3lem3b  27132  cdj3i  27133
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