Theorem chssii 29593

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 29589	. 2 ⊢ 𝐻 ∈ Sℋ
3	2	shssii 29575	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2106   ⊆ wss 3887   ℋchba 29281   C_ℋ cch 29291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-hilex 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441  df-ov 7278  df-sh 29569  df-ch 29583

This theorem is referenced by:  cheli  29594  chelii  29595  hhsscms  29640  chocvali  29661  chm1i  29818  chsscon3i  29823  chsscon2i  29825  chjoi  29850  chj1i  29851  shjshsi  29854  sshhococi  29908  h1dei  29912  spansnpji  29940  spanunsni  29941  h1datomi  29943  spansnji  30008  pjfi  30066  riesz3i  30424  hmopidmpji  30514  pjoccoi  30540  pjinvari  30553  stcltr2i  30637  mdsymi  30773  mdcompli  30791  dmdcompli  30792