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Theorem chne0i 29566
Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1 𝐴C
Assertion
Ref Expression
chne0i (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Distinct variable group:   𝑥,𝐴

Proof of Theorem chne0i
StepHypRef Expression
1 ch0le.1 . . 3 𝐴C
21chshii 29340 . 2 𝐴S
32shne0i 29561 1 (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2112  wne 2943  wrex 3065  0c0v 29037   C cch 29042  0c0h 29048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5209  ax-hilex 29112  ax-hv0cl 29116
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5575  df-cnv 5577  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-iota 6359  df-fv 6409  df-ov 7238  df-sh 29320  df-ch 29334  df-ch0 29366
This theorem is referenced by:  chne0  29607  hne0  29660  h1datomi  29694  riesz3i  30175  pjnmopi  30261
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