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Theorem lssne0 20222
Description: A nonzero subspace has a nonzero vector. (shne0i 29818 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 20212 . . . 4 (𝑋𝑆𝑋 ≠ ∅)
3 eqsn 4762 . . . 4 (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
42, 3syl 17 . . 3 (𝑋𝑆 → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
5 nne 2947 . . . . 5 𝑦0𝑦 = 0 )
65ralbii 3091 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ∀𝑦𝑋 𝑦 = 0 )
7 ralnex 3165 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
86, 7bitr3i 276 . . 3 (∀𝑦𝑋 𝑦 = 0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
94, 8bitr2di 288 . 2 (𝑋𝑆 → (¬ ∃𝑦𝑋 𝑦0𝑋 = { 0 }))
109necon1abid 2982 1 (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065  c0 4256  {csn 4561  cfv 6426  0gc0g 17160  LSubSpclss 20203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-lss 20204
This theorem is referenced by:  lsmsat  37030  lssatomic  37033  dochsatshpb  39474  hgmapvvlem3  39947
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