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Theorem shss 25900
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 25898 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 460 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simpld 459 1  |-  ( H  e.  SH  ->  H  C_ 
~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767    C_ wss 3476    X. cxp 4997   "cima 5002   CCcc 9491   ~Hchil 25609    +h cva 25610    .h csm 25611   0hc0v 25614   SHcsh 25618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-hilex 25689
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-sh 25897
This theorem is referenced by:  shel  25901  shex  25902  shssii  25903  shsubcl  25911  chss  25920  shsspwh  25937  hhsssh  25958  shocel  25973  shocsh  25975  ocss  25976  shocss  25977  shocorth  25983  shococss  25985  shorth  25986  shoccl  25996  shsel  26005  shintcli  26020  spanid  26038  shjval  26042  shjcl  26047  shlej1  26051  shlub  26105  chscllem2  26329  chscllem4  26331
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