Theorem sheli 28071

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
shssi.1	|- H e. SH

Assertion

Ref	Expression
sheli	|- ( A e. H -> A e. ~H )

Proof of Theorem sheli

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 |- H e. SH
2	1	shssii 28070	. 2 |- H C_ ~H
3	2	sseli 3599	1 |- ( A e. H -> A e. ~H )

Syntax hints:   -> wi 4   e. wcel 1990   ~Hchil 27776   SHcsh 27785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-sh 28064

This theorem is referenced by:  norm1exi  28107  hhssabloi  28119  hhssnv  28121  shscli  28176  shunssi  28227  shmodsi  28248  omlsii  28262  5oalem1  28513  5oalem2  28514  5oalem3  28515  5oalem5  28517  imaelshi  28917  pjimai  29035  shatomici  29217  shatomistici  29220  cdjreui  29291  cdj1i  29292  cdj3lem1  29293  cdj3lem2b  29296  cdj3lem3  29297  cdj3lem3b  29299  cdj3i  29300