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Theorem n0lplig 27678
Description: There is no "empty line" in a planar incidence geometry. (Contributed by AV, 28-Nov-2021.) (Proof shortened by BJ, 2-Dec-2021.)
Assertion
Ref Expression
n0lplig (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Proof of Theorem n0lplig
StepHypRef Expression
1 nsnlplig 27676 . 2 (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2 elirr 8656 . . . . 5 ¬ V ∈ V
3 snprc 4386 . . . . 5 (¬ V ∈ V ↔ {V} = ∅)
42, 3mpbi 220 . . . 4 {V} = ∅
54eqcomi 2778 . . 3 ∅ = {V}
65eleq1i 2839 . 2 (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
71, 6sylnibr 318 1 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1629  wcel 2143  Vcvv 3348  c0 4060  {csn 4313  Pligcplig 27669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033  ax-reg 8651
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-v 3350  df-dif 3723  df-un 3725  df-nul 4061  df-sn 4314  df-pr 4316  df-uni 4572  df-plig 27670
This theorem is referenced by:  pliguhgr  27681
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