Theorem chel 24784

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

Step	Hyp	Ref	Expression
1		chss 24783	. 2 |- ( H e. CH -> H C_ ~H )
2	1	sselda 3463	1 |- ( ( H e. CH /\ A e. H ) -> A e. ~H )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1758   ~Hchil 24472   CHcch 24482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-hilex 24552

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-xp 4953  df-cnv 4955  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fv 5533  df-ov 6202  df-sh 24760  df-ch 24775

This theorem is referenced by:  pjhtheu2  24970  pjspansn  25131  pjid  25249  atom1d  25908  sumdmdii  25970