Theorem n0lplig 30772

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

Step	Hyp	Ref	Expression
1		nsnlplig 30770	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5282	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4685	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 233	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2778	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2860	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 332	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {csn 4591  Pligcplig 30763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-v 3465  df-dif 3916  df-ss 3930  df-nul 4295  df-sn 4592  df-uni 4874  df-plig 30764

This theorem is referenced by:  pliguhgr  30775