Theorem chssii 29010

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 29006	. 2 ⊢ 𝐻 ∈ Sℋ
3	2	shssii 28992	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2114   ⊆ wss 3938   ℋchba 28698   C_ℋ cch 28708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-hilex 28778

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fv 6365  df-ov 7161  df-sh 28986  df-ch 29000

This theorem is referenced by:  cheli  29011  chelii  29012  hhsscms  29057  chocvali  29078  chm1i  29235  chsscon3i  29240  chsscon2i  29242  chjoi  29267  chj1i  29268  shjshsi  29271  sshhococi  29325  h1dei  29329  spansnpji  29357  spanunsni  29358  h1datomi  29360  spansnji  29425  pjfi  29483  riesz3i  29841  hmopidmpji  29931  pjoccoi  29957  pjinvari  29970  stcltr2i  30054  mdsymi  30190  mdcompli  30208  dmdcompli  30209