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Theorem shssii 30466
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 30463 . 2 (𝐻S𝐻 ⊆ ℋ)
31, 2ax-mp 5 1 𝐻 ⊆ ℋ
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wss 3949  chba 30172   S csh 30181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30252
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-sh 30460
This theorem is referenced by:  sheli  30467  shelii  30468  chssii  30484  hhssabloilem  30514  hhssabloi  30515  hhssnv  30517  hhssba  30524  shunssji  30622  shsval3i  30641  shjshsi  30745  span0  30795  spanuni  30797  imaelshi  31311  nlelchi  31314  hmopidmchi  31404  pjimai  31429  shatomistici  31614
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