Theorem shssii 30497

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
shssi.1	⊢ 𝐻 ∈ Sℋ

Assertion

Ref	Expression
shssii	⊢ 𝐻 ⊆ ℋ

Proof of Theorem shssii

Step	Hyp	Ref	Expression
1		shssi.1	. 2 ⊢ 𝐻 ∈ Sℋ
2		shss 30494	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
3	1, 2	ax-mp 5	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2107   ⊆ wss 3949   ℋchba 30203   S_ℋ csh 30212

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-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-sh 30491

This theorem is referenced by:  sheli  30498  shelii  30499  chssii  30515  hhssabloilem  30545  hhssabloi  30546  hhssnv  30548  hhssba  30555  shunssji  30653  shsval3i  30672  shjshsi  30776  span0  30826  spanuni  30828  imaelshi  31342  nlelchi  31345  hmopidmchi  31435  pjimai  31460  shatomistici  31645