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Theorem chne0i 28161
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 27933 . 2 𝐴S
32shne0i 28156 1 (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  wne 2790  wrex 2908  0c0v 27630   C cch 27635  0c0h 27641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-hilex 27705  ax-hv0cl 27709
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fv 5855  df-ov 6607  df-sh 27913  df-ch 27927  df-ch0 27959
This theorem is referenced by:  chne0  28202  hne0  28255  h1datomi  28289  riesz3i  28770  pjnmopi  28856
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