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Theorem shel 21790
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shel  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )

Proof of Theorem shel
StepHypRef Expression
1 shss 21789 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
21sselda 3180 1  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   ~Hchil 21499   SHcsh 21508
This theorem is referenced by:  shuni  21879  shsel3  21894  shscom  21898  shsel1  21900  elspancl  21916  pjpjpre  21998  spansnss  22150  sh1dle  22931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-sh 21786
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