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Theorem n0lplig 29467
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 29465 . 2 (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2 vprc 5273 . . . . 5 ¬ V ∈ V
3 snprc 4679 . . . . 5 (¬ V ∈ V ↔ {V} = ∅)
42, 3mpbi 229 . . . 4 {V} = ∅
54eqcomi 2742 . . 3 ∅ = {V}
65eleq1i 2825 . 2 (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
71, 6sylnibr 329 1 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  c0 4283  {csn 4587  Pligcplig 29458
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-ext 2704  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284  df-sn 4588  df-uni 4867  df-plig 29459
This theorem is referenced by:  pliguhgr  29470
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