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Theorem lssne0 20967
Description: A nonzero subspace has a nonzero vector. (shne0i 31477 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lss0cl.z 0 = (0g𝑊)
lss0cl.s 𝑆 = (LSubSp‘𝑊)
Assertion
Ref Expression
lssne0 (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
Distinct variable groups:   𝑦,𝑋   𝑦, 0
Allowed substitution hints:   𝑆(𝑦)   𝑊(𝑦)

Proof of Theorem lssne0
StepHypRef Expression
1 lss0cl.s . . . . 5 𝑆 = (LSubSp‘𝑊)
21lssn0 20956 . . . 4 (𝑋𝑆𝑋 ≠ ∅)
3 eqsn 4834 . . . 4 (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
42, 3syl 17 . . 3 (𝑋𝑆 → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
5 nne 2942 . . . . 5 𝑦0𝑦 = 0 )
65ralbii 3091 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ∀𝑦𝑋 𝑦 = 0 )
7 ralnex 3070 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
86, 7bitr3i 277 . . 3 (∀𝑦𝑋 𝑦 = 0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
94, 8bitr2di 288 . 2 (𝑋𝑆 → (¬ ∃𝑦𝑋 𝑦0𝑋 = { 0 }))
109necon1abid 2977 1 (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  c0 4339  {csn 4631  cfv 6563  0gc0g 17486  LSubSpclss 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-lss 20948
This theorem is referenced by:  lsmsat  38990  lssatomic  38993  dochsatshpb  41435  hgmapvvlem3  41908
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