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Theorem shne0i 28638
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 2934 . 2 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2 df-rex 3057 . . 3 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
3 nss 3805 . . 3 𝐴 ⊆ 0 ↔ ∃𝑥(𝑥𝐴 ∧ ¬ 𝑥 ∈ 0))
4 shne0.1 . . . . 5 𝐴S
5 shle0 28632 . . . . 5 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
64, 5ax-mp 5 . . . 4 (𝐴 ⊆ 0𝐴 = 0)
76notbii 309 . . 3 𝐴 ⊆ 0 ↔ ¬ 𝐴 = 0)
82, 3, 73bitr2ri 289 . 2 𝐴 = 0 ↔ ∃𝑥𝐴 ¬ 𝑥 ∈ 0)
9 elch0 28442 . . . 4 (𝑥 ∈ 0𝑥 = 0)
109necon3bbii 2980 . . 3 𝑥 ∈ 0𝑥 ≠ 0)
1110rexbii 3180 . 2 (∃𝑥𝐴 ¬ 𝑥 ∈ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
121, 8, 113bitri 286 1 (𝐴 ≠ 0 ↔ ∃𝑥𝐴 𝑥 ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2140  wne 2933  wrex 3052  wss 3716  0c0v 28112   S csh 28116  0c0h 28123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-hilex 28187  ax-hv0cl 28191
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-sh 28395  df-ch0 28441
This theorem is referenced by:  chne0i  28643  shatomici  29548
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