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Theorem shne0i 31271
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 2938 . 2 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2 df-rex 3068 . . 3 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
3 nss 4044 . . 3 𝐴 ⊆ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
4 shne0.1 . . . . 5 𝐴S
5 shle0 31265 . . . . 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 31077 . . . 4 (𝑥 ∈ 0𝑥 = 0)
109necon3bbii 2985 . . 3 𝑥 ∈ 0𝑥 ≠ 0)
1110rexbii 3091 . 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 205  wa 395   = wceq 1534  wex 1774  wcel 2099  wne 2937  wrex 3067  wss 3947  0c0v 30747   S csh 30751  0c0h 30758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-hilex 30822  ax-hv0cl 30826
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-sh 31030  df-ch0 31076
This theorem is referenced by:  chne0i  31276  shatomici  32181
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