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Theorem shne0i 22082
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 2481 . 2  |-  ( A  =/=  0H  <->  -.  A  =  0H )
2 df-rex 2583 . . 3  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
3 nss 3270 . . 3  |-  ( -.  A  C_  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
4 shne0.1 . . . . 5  |-  A  e.  SH
5 shle0 22076 . . . . 5  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
64, 5ax-mp 8 . . . 4  |-  ( A 
C_  0H  <->  A  =  0H )
76notbii 287 . . 3  |-  ( -.  A  C_  0H  <->  -.  A  =  0H )
82, 3, 73bitr2ri 265 . 2  |-  ( -.  A  =  0H  <->  E. x  e.  A  -.  x  e.  0H )
9 elch0 21888 . . . 4  |-  ( x  e.  0H  <->  x  =  0h )
109necon3bbii 2510 . . 3  |-  ( -.  x  e.  0H  <->  x  =/=  0h )
1110rexbii 2602 . 2  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x  e.  A  x  =/=  0h )
121, 8, 113bitri 262 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578    C_ wss 3186   0hc0v 21559   SHcsh 21563   0Hc0h 21570
This theorem is referenced by:  chne0i  22087  shatomici  22993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-hilex 21634  ax-hv0cl 21638
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-sh 21841  df-ch0 21887
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