Theorem shssi 9081

Description: A closed subspace of a Hilbert space is a subset of Hilbert space.

Hypothesis

Ref	Expression
shssi.1	|- H e. SH

Assertion

Ref	Expression
shssi	|- H (_ H~

Proof of Theorem shssi

Step	Hyp	Ref	Expression
1		shssi.1	. 2 |- H e. SH
2		shss 9079	. 2 |- (H e. SH -> H (_ H~)
3	1, 2	ax-mp 7	1 |- H (_ H~

Syntax hints:   e. wcel 958   (_ wss 2047  H~chil 8788  SHcsh 8797

This theorem is referenced by:  shel 9082  sheli 9083  chssi 9101  hhssabl 9132  hhssnv 9134  hhssba 9141  shslej 9338  shlub 9346  shsumval3 9361  shjshs 9415  span0 9465  spanun 9467  osum 9586  pjima 10104  shatomistic 10288

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-sh 9076

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