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Theorem cheli 25923
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 25922 . 2  |-  H  C_  ~H
32sseli 3500 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ~Hchil 25609   CHcch 25619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fv 5596  df-ov 6288  df-sh 25897  df-ch 25912
This theorem is referenced by:  pjhthlem1  26082  pjhthlem2  26083  h1de2ci  26247  spanunsni  26270  spansncvi  26343  3oalem1  26353  pjcompi  26363  pjocini  26389  pjjsi  26391  pjrni  26393  pjdsi  26403  pjds3i  26404  mayete3i  26419  mayete3iOLD  26420  riesz3i  26754  pjnmopi  26840  pjnormssi  26860  pjimai  26868  pjclem4a  26890  pjclem4  26891  pj3lem1  26898  pj3si  26899  strlem1  26942  strlem3  26945  strlem5  26947  hstrlem3  26953  hstrlem5  26955  sumdmdii  27107  sumdmdlem  27110  sumdmdlem2  27111
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