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Theorem cheli 21828
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 21827 . 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   CHcch 21525
This theorem is referenced by:  pjhthlem1  21986  pjhthlem2  21987  h1de2ci  22151  spanunsni  22174  spansncvi  22247  3oalem1  22257  pjcompi  22267  pjocini  22293  pjjsi  22295  pjrni  22297  pjdsi  22307  pjds3i  22308  mayete3i  22323  mayete3iOLD  22324  riesz3i  22658  pjnmopi  22744  pjnormssi  22764  pjimai  22772  pjclem4a  22794  pjclem4  22795  pj3lem1  22802  pj3si  22803  strlem1  22846  strlem3  22849  strlem5  22851  hstrlem3  22857  hstrlem5  22859  sumdmdii  23011  sumdmdlem  23014  sumdmdlem2  23015
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-rex 2562  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-uni 3844  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-iota 5235  df-fv 5279  df-ov 5877  df-sh 21802  df-ch 21817
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