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Theorem shelii 22558
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
shssi.1  |-  H  e.  SH
sheli.1  |-  A  e.  H
Assertion
Ref Expression
shelii  |-  A  e. 
~H

Proof of Theorem shelii
StepHypRef Expression
1 shssi.1 . . 3  |-  H  e.  SH
21shssii 22556 . 2  |-  H  C_  ~H
3 sheli.1 . 2  |-  A  e.  H
42, 3sselii 3281 1  |-  A  e. 
~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   ~Hchil 22263   SHcsh 22272
This theorem is referenced by:  omlsilem  22745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-sh 22550
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