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

Step	Hyp	Ref	Expression
1		chss 24583	. 2 |- ( H e. CH -> H C_ ~H )
2	1	sselda 3351	1 |- ( ( H e. CH /\ A e. H ) -> A e. ~H )

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