Theorem chssii 28802

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 ∈ Cℋ

Assertion

Ref	Expression
chssii	⊢ 𝐻 ⊆ ℋ

Proof of Theorem chssii

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 ⊢ 𝐻 ∈ Cℋ
2	1	chshii 28798	. 2 ⊢ 𝐻 ∈ Sℋ
3	2	shssii 28784	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2051   ⊆ wss 3822   ℋchba 28490   C_ℋ cch 28500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-hilex 28570

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-xp 5409  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fv 6193  df-ov 6977  df-sh 28778  df-ch 28792

This theorem is referenced by:  cheli  28803  chelii  28804  hhsscms  28850  chocvali  28872  chm1i  29029  chsscon3i  29034  chsscon2i  29036  chjoi  29061  chj1i  29062  shjshsi  29065  sshhococi  29119  h1dei  29123  spansnpji  29151  spanunsni  29152  h1datomi  29154  spansnji  29219  pjfi  29277  riesz3i  29635  hmopidmpji  29725  pjoccoi  29751  pjinvari  29764  stcltr2i  29848  mdsymi  29984  mdcompli  30002  dmdcompli  30003