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Theorem cheli 21812
Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chssi.1  |-  H  e. 
CH
Assertion
Ref Expression
cheli  |-  ( A  e.  H  ->  A  e.  ~H )

Proof of Theorem cheli
StepHypRef Expression
1 chssi.1 . . 3  |-  H  e. 
CH
21chssii 21811 . 2  |-  H  C_  ~H
32sseli 3176 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   ~Hchil 21499   CHcch 21509
This theorem is referenced by:  pjhthlem1  21970  pjhthlem2  21971  h1de2ci  22135  spanunsni  22158  spansncvi  22231  3oalem1  22241  pjcompi  22251  pjocini  22277  pjjsi  22279  pjrni  22281  pjdsi  22291  pjds3i  22292  mayete3i  22307  mayete3iOLD  22308  riesz3i  22642  pjnmopi  22728  pjnormssi  22748  pjimai  22756  pjclem4a  22778  pjclem4  22779  pj3lem1  22786  pj3si  22787  strlem1  22830  strlem3  22833  strlem5  22835  hstrlem3  22841  hstrlem5  22843  sumdmdii  22995  sumdmdlem  22998  sumdmdlem2  22999
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-rex 2549  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-uni 3828  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-iota 5219  df-fv 5263  df-ov 5861  df-sh 21786  df-ch 21801
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