Theorem n0lplig 28746

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 28744	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5234	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4650	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 229	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2747	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2829	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 328	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  {csn 4558  Pligcplig 28737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-uni 4837  df-plig 28738

This theorem is referenced by:  pliguhgr  28749