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Theorem shne0i 22950
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 2601 . 2  |-  ( A  =/=  0H  <->  -.  A  =  0H )
2 df-rex 2711 . . 3  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
3 nss 3406 . . 3  |-  ( -.  A  C_  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
4 shne0.1 . . . . 5  |-  A  e.  SH
5 shle0 22944 . . . . 5  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
64, 5ax-mp 8 . . . 4  |-  ( A 
C_  0H  <->  A  =  0H )
76notbii 288 . . 3  |-  ( -.  A  C_  0H  <->  -.  A  =  0H )
82, 3, 73bitr2ri 266 . 2  |-  ( -.  A  =  0H  <->  E. x  e.  A  -.  x  e.  0H )
9 elch0 22756 . . . 4  |-  ( x  e.  0H  <->  x  =  0h )
109necon3bbii 2632 . . 3  |-  ( -.  x  e.  0H  <->  x  =/=  0h )
1110rexbii 2730 . 2  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x  e.  A  x  =/=  0h )
121, 8, 113bitri 263 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320   0hc0v 22427   SHcsh 22431   0Hc0h 22438
This theorem is referenced by:  chne0i  22955  shatomici  23861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-hilex 22502  ax-hv0cl 22506
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-sh 22709  df-ch0 22755
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