Theorem n0lplig 28841

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 28839	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5243	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4659	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 229	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2749	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2831	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 329	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2110  Vcvv 3431  ∅c0 4262  {csn 4567  Pligcplig 28832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-v 3433  df-dif 3895  df-in 3899  df-ss 3909  df-nul 4263  df-sn 4568  df-uni 4846  df-plig 28833

This theorem is referenced by:  pliguhgr  28844