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Theorem lssne0 17792
Description: A nonzero subspace has a nonzero vector. (shne0i 26564 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 17782 . . . 4  |-  ( X  e.  S  ->  X  =/=  (/) )
3 eqsn 4177 . . . 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 2655 . . . . 5  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
65ralbii 2885 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  A. y  e.  X  y  =  .0.  )
7 ralnex 2900 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
86, 7bitr3i 251 . . 3  |-  ( A. y  e.  X  y  =  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
94, 8syl6rbb 262 . 2  |-  ( X  e.  S  ->  ( -.  E. y  e.  X  y  =/=  .0.  <->  X  =  {  .0.  } ) )
109necon1abid 2702 1  |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   (/)c0 3783   {csn 4016   ` cfv 5570   0gc0g 14929   LSubSpclss 17773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-lss 17774
This theorem is referenced by:  lsmsat  35130  lssatomic  35133  dochsatshpb  37576  hgmapvvlem3  38052
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