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Theorem shne0i 27043
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 2720 . . 3  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
3 nss 3465 . . 3  |-  ( -.  A  C_  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
4 shne0.1 . . . . 5  |-  A  e.  SH
5 shle0 27037 . . . . 5  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
64, 5ax-mp 5 . . . 4  |-  ( A 
C_  0H  <->  A  =  0H )
76notbii 297 . . 3  |-  ( -.  A  C_  0H  <->  -.  A  =  0H )
82, 3, 73bitr2ri 277 . 2  |-  ( -.  A  =  0H  <->  E. x  e.  A  -.  x  e.  0H )
9 elch0 26849 . . . 4  |-  ( x  e.  0H  <->  x  =  0h )
109necon3bbii 2648 . . 3  |-  ( -.  x  e.  0H  <->  x  =/=  0h )
1110rexbii 2866 . 2  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x  e.  A  x  =/=  0h )
121, 8, 113bitri 274 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   E.wrex 2715    C_ wss 3379   0hc0v 26519   SHcsh 26523   0Hc0h 26530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-hilex 26594  ax-hv0cl 26598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802  df-cnv 4804  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-sh 26802  df-ch0 26848
This theorem is referenced by:  chne0i  27048  shatomici  27953
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