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Theorem lssne0 20835
Description: A nonzero subspace has a nonzero vector. (shne0i 31271 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 20824 . . . 4 (𝑋𝑆𝑋 ≠ ∅)
3 eqsn 4833 . . . 4 (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
42, 3syl 17 . . 3 (𝑋𝑆 → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
5 nne 2941 . . . . 5 𝑦0𝑦 = 0 )
65ralbii 3090 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ∀𝑦𝑋 𝑦 = 0 )
7 ralnex 3069 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
86, 7bitr3i 277 . . 3 (∀𝑦𝑋 𝑦 = 0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
94, 8bitr2di 288 . 2 (𝑋𝑆 → (¬ ∃𝑦𝑋 𝑦0𝑋 = { 0 }))
109necon1abid 2976 1 (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1534  wcel 2099  wne 2937  wral 3058  wrex 3067  c0 4323  {csn 4629  cfv 6548  0gc0g 17421  LSubSpclss 20815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-lss 20816
This theorem is referenced by:  lsmsat  38480  lssatomic  38483  dochsatshpb  40925  hgmapvvlem3  41398
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