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Theorem lssne0 20426
Description: A nonzero subspace has a nonzero vector. (shne0i 30432 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 20416 . . . 4 (𝑋𝑆𝑋 ≠ ∅)
3 eqsn 4790 . . . 4 (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
42, 3syl 17 . . 3 (𝑋𝑆 → (𝑋 = { 0 } ↔ ∀𝑦𝑋 𝑦 = 0 ))
5 nne 2944 . . . . 5 𝑦0𝑦 = 0 )
65ralbii 3093 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ∀𝑦𝑋 𝑦 = 0 )
7 ralnex 3072 . . . 4 (∀𝑦𝑋 ¬ 𝑦0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
86, 7bitr3i 277 . . 3 (∀𝑦𝑋 𝑦 = 0 ↔ ¬ ∃𝑦𝑋 𝑦0 )
94, 8bitr2di 288 . 2 (𝑋𝑆 → (¬ ∃𝑦𝑋 𝑦0𝑋 = { 0 }))
109necon1abid 2979 1 (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  c0 4283  {csn 4587  cfv 6497  0gc0g 17326  LSubSpclss 20407
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-lss 20408
This theorem is referenced by:  lsmsat  37516  lssatomic  37519  dochsatshpb  39961  hgmapvvlem3  40434
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