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Theorem lssne0 16028
Description: A nonzero subspace has a nonzero vector. (shne0i 22951 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lss0cl.z  |-  .0.  =  ( 0g `  W )
lss0cl.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssne0  |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
Distinct variable groups:    y, X    y,  .0.
Allowed substitution hints:    S( y)    W( y)

Proof of Theorem lssne0
StepHypRef Expression
1 lss0cl.s . . . . 5  |-  S  =  ( LSubSp `  W )
21lssn0 16018 . . . 4  |-  ( X  e.  S  ->  X  =/=  (/) )
3 eqsn 3961 . . . 4  |-  ( X  =/=  (/)  ->  ( X  =  {  .0.  }  <->  A. y  e.  X  y  =  .0.  ) )
42, 3syl 16 . . 3  |-  ( X  e.  S  ->  ( X  =  {  .0.  }  <->  A. y  e.  X  y  =  .0.  )
)
5 nne 2606 . . . . 5  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
65ralbii 2730 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  A. y  e.  X  y  =  .0.  )
7 ralnex 2716 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
86, 7bitr3i 244 . . 3  |-  ( A. y  e.  X  y  =  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
94, 8syl6rbb 255 . 2  |-  ( X  e.  S  ->  ( -.  E. y  e.  X  y  =/=  .0.  <->  X  =  {  .0.  } ) )
109necon1abid 2658 1  |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   (/)c0 3629   {csn 3815   ` cfv 5455   0gc0g 13724   LSubSpclss 16009
This theorem is referenced by:  lsmsat  29807  lssatomic  29810  dochsatshpb  32251  hgmapvvlem3  32727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-lss 16010
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