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Theorem shne0i 31467
Description: A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
shne0.1 𝐴S
Assertion
Ref Expression
shne0i (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Distinct variable group:   𝑥,𝐴

Proof of Theorem shne0i
StepHypRef Expression
1 df-ne 2941 . 2 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2 df-rex 3071 . . 3 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
3 nss 4048 . . 3 𝐴 ⊆ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
4 shne0.1 . . . . 5 𝐴S
5 shle0 31461 . . . . 5 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
64, 5ax-mp 5 . . . 4 (𝐴 ⊆ 0𝐴 = 0)
76notbii 320 . . 3 𝐴 ⊆ 0 ↔ ¬ 𝐴 = 0)
82, 3, 73bitr2ri 300 . 2 𝐴 = 0 ↔ ∃𝑥𝐴 ¬ 𝑥 ∈ 0)
9 elch0 31273 . . . 4 (𝑥 ∈ 0𝑥 = 0)
109necon3bbii 2988 . . 3 𝑥 ∈ 0𝑥 ≠ 0)
1110rexbii 3094 . 2 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
121, 8, 113bitri 297 1 (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  wss 3951  0c0v 30943   S csh 30947  0c0h 30954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-hilex 31018  ax-hv0cl 31022
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-sh 31226  df-ch0 31272
This theorem is referenced by:  chne0i  31472  shatomici  32377
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