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Theorem sheli 21809
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 21808 . 2  |-  H  C_  ~H
32sseli 3189 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   ~Hchil 21515   SHcsh 21524
This theorem is referenced by:  norm1exi  21845  hhssabloi  21855  hhssnv  21857  shscli  21912  shunssi  21963  shmodsi  21984  omlsii  21998  5oalem1  22249  5oalem2  22250  5oalem3  22251  5oalem5  22253  imaelshi  22654  pjimai  22772  shatomici  22954  shatomistici  22957  cdjreui  23028  cdj1i  23029  cdj3lem1  23030  cdj3lem2b  23033  cdj3lem3  23034  cdj3lem3b  23036  cdj3i  23037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-sh 21802
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