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Theorem sheli 26743
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 26742 . 2  |-  H  C_  ~H
32sseli 3457 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   ~Hchil 26448   SHcsh 26457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-hilex 26528
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-sh 26736
This theorem is referenced by:  norm1exi  26779  hhssabloi  26789  hhssnv  26791  shscli  26846  shunssi  26897  shmodsi  26918  omlsii  26932  5oalem1  27183  5oalem2  27184  5oalem3  27185  5oalem5  27187  imaelshi  27587  pjimai  27705  shatomici  27887  shatomistici  27890  cdjreui  27961  cdj1i  27962  cdj3lem1  27963  cdj3lem2b  27966  cdj3lem3  27967  cdj3lem3b  27969  cdj3i  27970
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