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Theorem shel 21806
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 21805 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
21sselda 3193 1  |-  ( ( H  e.  SH  /\  A  e.  H )  ->  A  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   ~Hchil 21515   SHcsh 21524
This theorem is referenced by:  shuni  21895  shsel3  21910  shscom  21914  shsel1  21916  elspancl  21932  pjpjpre  22014  spansnss  22166  sh1dle  22947
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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