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Theorem chne0i 26035
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 25809 . 2  |-  A  e.  SH
32shne0i 26030 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1762    =/= wne 2657   E.wrex 2810   0hc0v 25505   CHcch 25510   0Hc0h 25516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-hilex 25580  ax-hv0cl 25584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fv 5589  df-ov 6280  df-sh 25788  df-ch 25803  df-ch0 25835
This theorem is referenced by:  chne0  26076  hne0  26129  h1datomi  26163  riesz3i  26645  pjnmopi  26731
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