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Theorem shssii 28917
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 28914 . 2 (𝐻S𝐻 ⊆ ℋ)
31, 2ax-mp 5 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  wss 3933  chba 28623   S csh 28632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-sh 28911
This theorem is referenced by:  sheli  28918  shelii  28919  chssii  28935  hhssabloilem  28965  hhssabloi  28966  hhssnv  28968  hhssba  28975  shunssji  29073  shsval3i  29092  shjshsi  29196  span0  29246  spanuni  29248  imaelshi  29762  nlelchi  29765  hmopidmchi  29855  pjimai  29880  shatomistici  30065
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