Theorem chssii 30479

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 30475	. 2 ⊢ 𝐻 ∈ Sℋ
3	2	shssii 30461	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2106   ⊆ wss 3948   ℋchba 30167   C_ℋ cch 30177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-hilex 30247

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-ov 7411  df-sh 30455  df-ch 30469

This theorem is referenced by:  cheli  30480  chelii  30481  hhsscms  30526  chocvali  30547  chm1i  30704  chsscon3i  30709  chsscon2i  30711  chjoi  30736  chj1i  30737  shjshsi  30740  sshhococi  30794  h1dei  30798  spansnpji  30826  spanunsni  30827  h1datomi  30829  spansnji  30894  pjfi  30952  riesz3i  31310  hmopidmpji  31400  pjoccoi  31426  pjinvari  31439  stcltr2i  31523  mdsymi  31659  mdcompli  31677  dmdcompli  31678