Theorem shssii 29476

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 29473	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
3	1, 2	ax-mp 5	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2108   ⊆ wss 3883   ℋchba 29182   S_ℋ csh 29191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-sh 29470

This theorem is referenced by:  sheli  29477  shelii  29478  chssii  29494  hhssabloilem  29524  hhssabloi  29525  hhssnv  29527  hhssba  29534  shunssji  29632  shsval3i  29651  shjshsi  29755  span0  29805  spanuni  29807  imaelshi  30321  nlelchi  30324  hmopidmchi  30414  pjimai  30439  shatomistici  30624