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Theorem lssne0 18772
 Description: A nonzero subspace has a nonzero vector. (shne0i 27691 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 18762 . . . 4 (𝑋𝑆𝑋 ≠ ∅)
3 eqsn 4301 . . . 4 (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
42, 3syl 17 . . 3 (𝑋𝑆 → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
5 nne 2786 . . . . 5 𝑦0𝑦 = 0 )
65ralbii 2963 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ∀𝑦𝑋 𝑦 = 0 )
7 ralnex 2975 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
86, 7bitr3i 265 . . 3 (∀𝑦𝑋 𝑦 = 0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
94, 8syl6rbb 276 . 2 (𝑋𝑆 → (¬ ∃𝑦𝑋 𝑦0𝑋 = { 0 }))
109necon1abid 2820 1 (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∅c0 3874  {csn 4125  ‘cfv 5804  0gc0g 15923  LSubSpclss 18753 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-lss 18754 This theorem is referenced by:  lsmsat  33313  lssatomic  33316  dochsatshpb  35759  hgmapvvlem3  36235
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