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Theorem shssii 22715
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  e.  SH
Assertion
Ref Expression
shssii  |-  H  C_  ~H

Proof of Theorem shssii
StepHypRef Expression
1 shssi.1 . 2  |-  H  e.  SH
2 shss 22712 . 2  |-  ( H  e.  SH  ->  H  C_ 
~H )
31, 2ax-mp 8 1  |-  H  C_  ~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1725    C_ wss 3320   ~Hchil 22422   SHcsh 22431
This theorem is referenced by:  sheli  22716  shelii  22717  chssii  22734  hhssabloi  22762  hhssnv  22764  hhssba  22771  shunssji  22871  shsval3i  22890  shjshsi  22994  span0  23044  spanuni  23046  imaelshi  23561  nlelchi  23564  hmopidmchi  23654  pjimai  23679  shatomistici  23864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-sh 22709
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