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Theorem sheli 22709
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 22708 . 2  |-  H  C_  ~H
32sseli 3337 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ~Hchil 22415   SHcsh 22424
This theorem is referenced by:  norm1exi  22745  hhssabloi  22755  hhssnv  22757  shscli  22812  shunssi  22863  shmodsi  22884  omlsii  22898  5oalem1  23149  5oalem2  23150  5oalem3  23151  5oalem5  23153  imaelshi  23554  pjimai  23672  shatomici  23854  shatomistici  23857  cdjreui  23928  cdj1i  23929  cdj3lem1  23930  cdj3lem2b  23933  cdj3lem3  23934  cdj3lem3b  23936  cdj3i  23937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-hilex 22495
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-cnv 4879  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-sh 22702
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