Theorem chel 26548

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 26547	. 2 |- ( H e. CH -> H C_ ~H )
2	1	sselda 3441	1 |- ( ( H e. CH /\ A e. H ) -> A e. ~H )

Syntax hints:   -> wi 4   /\ wa 367   e. wcel 1842   ~Hchil 26236   CHcch 26246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-hilex 26316

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fv 5576  df-ov 6280  df-sh 26524  df-ch 26539

This theorem is referenced by:  pjhtheu2  26734  pjspansn  26895  pjid  27013  atom1d  27671  sumdmdii  27733