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Theorem chel 24584
Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)
Assertion
Ref Expression
chel  |-  ( ( H  e.  CH  /\  A  e.  H )  ->  A  e.  ~H )

Proof of Theorem chel
StepHypRef Expression
1 chss 24583 . 2  |-  ( H  e.  CH  ->  H  C_ 
~H )
21sselda 3351 1  |-  ( ( H  e.  CH  /\  A  e.  H )  ->  A  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   ~Hchil 24272   CHcch 24282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-hilex 24352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fv 5421  df-ov 6089  df-sh 24560  df-ch 24575
This theorem is referenced by:  pjhtheu2  24770  pjspansn  24931  pjid  25049  atom1d  25708  sumdmdii  25770
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