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Theorem lssne0 15708
Description: A nonzero subspace has a nonzero vector. (shne0i 22027 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 15698 . . . 4  |-  ( X  e.  S  ->  X  =/=  (/) )
3 eqsn 3775 . . . 4  |-  ( X  =/=  (/)  ->  ( X  =  {  .0.  }  <->  A. y  e.  X  y  =  .0.  ) )
42, 3syl 15 . . 3  |-  ( X  e.  S  ->  ( X  =  {  .0.  }  <->  A. y  e.  X  y  =  .0.  )
)
5 nne 2450 . . . . 5  |-  ( -.  y  =/=  .0.  <->  y  =  .0.  )
65ralbii 2567 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  A. y  e.  X  y  =  .0.  )
7 ralnex 2553 . . . 4  |-  ( A. y  e.  X  -.  y  =/=  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
86, 7bitr3i 242 . . 3  |-  ( A. y  e.  X  y  =  .0.  <->  -.  E. y  e.  X  y  =/=  .0.  )
94, 8syl6rbb 253 . 2  |-  ( X  e.  S  ->  ( -.  E. y  e.  X  y  =/=  .0.  <->  X  =  {  .0.  } ) )
109necon1abid 2499 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 176    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   {csn 3640   ` cfv 5255   0gc0g 13400   LSubSpclss 15689
This theorem is referenced by:  lsmsat  29198  lssatomic  29201  dochsatshpb  31642  hgmapvvlem3  32118
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-lss 15690
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