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Theorem shss 21789
Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
shss  |-  ( H  e.  SH  ->  H  C_ 
~H )

Proof of Theorem shss
StepHypRef Expression
1 issh 21787 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 446 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simpld 445 1  |-  ( H  e.  SH  ->  H  C_ 
~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    C_ wss 3152    X. cxp 4687   "cima 4692   CCcc 8735   ~Hchil 21499    +h cva 21500    .h csm 21501   0hc0v 21504   SHcsh 21508
This theorem is referenced by:  shel  21790  shex  21791  shssii  21792  shsubcl  21800  chss  21809  shsspwh  21825  hhsssh  21846  shocel  21861  shocsh  21863  ocss  21864  shocss  21865  shocorth  21871  shococss  21873  shorth  21874  shoccl  21884  shsel  21893  shintcli  21908  spanid  21926  shjval  21930  shjcl  21935  shlej1  21939  shlub  21993  chscllem2  22217  chscllem4  22219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-sh 21786
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