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Theorem shne0i 22955
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 2603 . 2  |-  ( A  =/=  0H  <->  -.  A  =  0H )
2 df-rex 2713 . . 3  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
3 nss 3408 . . 3  |-  ( -.  A  C_  0H  <->  E. x
( x  e.  A  /\  -.  x  e.  0H ) )
4 shne0.1 . . . . 5  |-  A  e.  SH
5 shle0 22949 . . . . 5  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
64, 5ax-mp 5 . . . 4  |-  ( A 
C_  0H  <->  A  =  0H )
76notbii 289 . . 3  |-  ( -.  A  C_  0H  <->  -.  A  =  0H )
82, 3, 73bitr2ri 267 . 2  |-  ( -.  A  =  0H  <->  E. x  e.  A  -.  x  e.  0H )
9 elch0 22761 . . . 4  |-  ( x  e.  0H  <->  x  =  0h )
109necon3bbii 2634 . . 3  |-  ( -.  x  e.  0H  <->  x  =/=  0h )
1110rexbii 2732 . 2  |-  ( E. x  e.  A  -.  x  e.  0H  <->  E. x  e.  A  x  =/=  0h )
121, 8, 113bitri 264 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    C_ wss 3322   0hc0v 22432   SHcsh 22436   0Hc0h 22443
This theorem is referenced by:  chne0i  22960  shatomici  23866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-hilex 22507  ax-hv0cl 22511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-sh 22714  df-ch0 22760
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