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Theorem lssne0 17147
Description: A nonzero subspace has a nonzero vector. (shne0i 24996 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 17137 . . . 4  |-  ( X  e.  S  ->  X  =/=  (/) )
3 eqsn 4135 . . . 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 2650 . . . . 5  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
65ralbii 2834 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  A. y  e.  X  y  =  .0.  )
7 ralnex 2846 . . . 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 2696 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 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   (/)c0 3738   {csn 3978   ` cfv 5519   0gc0g 14489   LSubSpclss 17128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-lss 17129
This theorem is referenced by:  lsmsat  32962  lssatomic  32965  dochsatshpb  35406  hgmapvvlem3  35882
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