Theorem chssii 30515

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 30511	. 2 ⊢ 𝐻 ∈ Sℋ
3	2	shssii 30497	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2107   ⊆ wss 3949   ℋchba 30203   C_ℋ cch 30213

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30283

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-sh 30491  df-ch 30505

This theorem is referenced by:  cheli  30516  chelii  30517  hhsscms  30562  chocvali  30583  chm1i  30740  chsscon3i  30745  chsscon2i  30747  chjoi  30772  chj1i  30773  shjshsi  30776  sshhococi  30830  h1dei  30834  spansnpji  30862  spanunsni  30863  h1datomi  30865  spansnji  30930  pjfi  30988  riesz3i  31346  hmopidmpji  31436  pjoccoi  31462  pjinvari  31475  stcltr2i  31559  mdsymi  31695  mdcompli  31713  dmdcompli  31714