Theorem chssii 28088

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

Hypothesis

Ref	Expression
chssi.1	|- H e. CH

Assertion

Ref	Expression
chssii	|- H C_ ~H

Proof of Theorem chssii

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 |- H e. CH
2	1	chshii 28084	. 2 |- H e. SH
3	2	shssii 28070	1 |- H C_ ~H

Syntax hints:   e. wcel 1990   C_ wss 3574   ~Hchil 27776   CHcch 27786

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-rex 2918  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-uni 4437  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-iota 5851  df-fv 5896  df-ov 6653  df-sh 28064  df-ch 28078

This theorem is referenced by:  cheli  28089  chelii  28090  hhsscms  28136  chocvali  28158  chm1i  28315  chsscon3i  28320  chsscon2i  28322  chjoi  28347  chj1i  28348  shjshsi  28351  sshhococi  28405  h1dei  28409  spansnpji  28437  spanunsni  28438  h1datomi  28440  spansnji  28505  pjfi  28563  riesz3i  28921  hmopidmpji  29011  pjoccoi  29037  pjinvari  29050  stcltr2i  29134  mdsymi  29270  mdcompli  29288  dmdcompli  29289