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Theorem n0lplig 30240
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 30238 . 2 (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2 vprc 5308 . . . . 5 ¬ V ∈ V
3 snprc 4716 . . . . 5 (¬ V ∈ V ↔ {V} = ∅)
42, 3mpbi 229 . . . 4 {V} = ∅
54eqcomi 2735 . . 3 ∅ = {V}
65eleq1i 2818 . 2 (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
71, 6sylnibr 329 1 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  {csn 4623  Pligcplig 30231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-uni 4903  df-plig 30232
This theorem is referenced by:  pliguhgr  30243
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