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Theorem shne0i 29229
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 3012 . 2 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2 df-rex 3136 . . 3 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
3 nss 4004 . . 3 𝐴 ⊆ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
4 shne0.1 . . . . 5 𝐴S
5 shle0 29223 . . . . 5 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
64, 5ax-mp 5 . . . 4 (𝐴 ⊆ 0𝐴 = 0)
76notbii 323 . . 3 𝐴 ⊆ 0 ↔ ¬ 𝐴 = 0)
82, 3, 73bitr2ri 303 . 2 𝐴 = 0 ↔ ∃𝑥𝐴 ¬ 𝑥 ∈ 0)
9 elch0 29035 . . . 4 (𝑥 ∈ 0𝑥 = 0)
109necon3bbii 3058 . . 3 𝑥 ∈ 0𝑥 ≠ 0)
1110rexbii 3235 . 2 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
121, 8, 113bitri 300 1 (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2114  wne 3011  wrex 3131  wss 3908  0c0v 28705   S csh 28709  0c0h 28716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-hilex 28780  ax-hv0cl 28784
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-rex 3136  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-sh 28988  df-ch0 29034
This theorem is referenced by:  chne0i  29234  shatomici  30139
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