Theorem chel 25824

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 25823	. 2 |- ( H e. CH -> H C_ ~H )
2	1	sselda 3504	1 |- ( ( H e. CH /\ A e. H ) -> A e. ~H )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1767   ~Hchil 25512   CHcch 25522

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 25592

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 5549  df-fv 5594  df-ov 6285  df-sh 25800  df-ch 25815

This theorem is referenced by:  pjhtheu2  26010  pjspansn  26171  pjid  26289  atom1d  26948  sumdmdii  27010