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Theorem shne0i 24998
Description: A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
shne0.1  |-  A  e.  SH
Assertion
Ref Expression
shne0i  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Distinct variable group:    x, A

Proof of Theorem shne0i
StepHypRef Expression
1 df-ne 2647 . 2  |-  ( A  =/=  0H  <->  -.  A  =  0H )
2 df-rex 2802 . . 3  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
3 nss 3517 . . 3  |-  ( -.  A  C_  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
4 shne0.1 . . . . 5  |-  A  e.  SH
5 shle0 24992 . . . . 5  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
64, 5ax-mp 5 . . . 4  |-  ( A 
C_  0H  <->  A  =  0H )
76notbii 296 . . 3  |-  ( -.  A  C_  0H  <->  -.  A  =  0H )
82, 3, 73bitr2ri 274 . 2  |-  ( -.  A  =  0H  <->  E. x  e.  A  -.  x  e.  0H )
9 elch0 24804 . . . 4  |-  ( x  e.  0H  <->  x  =  0h )
109necon3bbii 2710 . . 3  |-  ( -.  x  e.  0H  <->  x  =/=  0h )
1110rexbii 2861 . 2  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x  e.  A  x  =/=  0h )
121, 8, 113bitri 271 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2645   E.wrex 2797    C_ wss 3431   0hc0v 24473   SHcsh 24477   0Hc0h 24484
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-hilex 24548  ax-hv0cl 24552
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-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-xp 4949  df-cnv 4951  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-sh 24756  df-ch0 24803
This theorem is referenced by:  chne0i  25003  shatomici  25909
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