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Theorem chne0i 26785
Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1  |-  A  e. 
CH
Assertion
Ref Expression
chne0i  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Distinct variable group:    x, A

Proof of Theorem chne0i
StepHypRef Expression
1 ch0le.1 . . 3  |-  A  e. 
CH
21chshii 26559 . 2  |-  A  e.  SH
32shne0i 26780 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1842    =/= wne 2598   E.wrex 2755   0hc0v 26255   CHcch 26260   0Hc0h 26266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-hilex 26330  ax-hv0cl 26334
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fv 5577  df-ov 6281  df-sh 26538  df-ch 26553  df-ch0 26585
This theorem is referenced by:  chne0  26826  hne0  26879  h1datomi  26913  riesz3i  27394  pjnmopi  27480
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