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Theorem shssii 28625
 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 𝐻S
Assertion
Ref Expression
shssii 𝐻 ⊆ ℋ

Proof of Theorem shssii
StepHypRef Expression
1 shssi.1 . 2 𝐻S
2 shss 28622 . 2 (𝐻S𝐻 ⊆ ℋ)
31, 2ax-mp 5 1 𝐻 ⊆ ℋ
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2166   ⊆ wss 3798   ℋchba 28331   Sℋ csh 28340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-hilex 28411 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-sh 28619 This theorem is referenced by:  sheli  28626  shelii  28627  chssii  28643  hhssabloilem  28673  hhssabloi  28674  hhssnv  28676  hhssba  28683  shunssji  28783  shsval3i  28802  shjshsi  28906  span0  28956  spanuni  28958  imaelshi  29472  nlelchi  29475  hmopidmchi  29565  pjimai  29590  shatomistici  29775
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