Theorem shssii 31299

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 31296	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
3	1, 2	ax-mp 5	1 ⊢ 𝐻 ⊆ ℋ

Syntax hints:   ∈ wcel 2114   ⊆ wss 3890   ℋchba 31005   S_ℋ csh 31014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-hilex 31085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-sh 31293

This theorem is referenced by:  sheli  31300  shelii  31301  chssii  31317  hhssabloilem  31347  hhssabloi  31348  hhssnv  31350  hhssba  31357  shunssji  31455  shsval3i  31474  shjshsi  31578  span0  31628  spanuni  31630  imaelshi  32144  nlelchi  32147  hmopidmchi  32237  pjimai  32262  shatomistici  32447