Theorem shssii 31245

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 31242	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
3	1, 2	ax-mp 5	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2108   ⊆ wss 3976   ℋchba 30951   S_ℋ csh 30960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-hilex 31031

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-sh 31239

This theorem is referenced by:  sheli  31246  shelii  31247  chssii  31263  hhssabloilem  31293  hhssabloi  31294  hhssnv  31296  hhssba  31303  shunssji  31401  shsval3i  31420  shjshsi  31524  span0  31574  spanuni  31576  imaelshi  32090  nlelchi  32093  hmopidmchi  32183  pjimai  32208  shatomistici  32393