Theorem n0lplig 30449

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 30447	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5297	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4699	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 230	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2743	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2824	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 329	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3464  ∅c0 4315  {csn 4608  Pligcplig 30440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5278

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-v 3466  df-dif 3936  df-ss 3950  df-nul 4316  df-sn 4609  df-uni 4890  df-plig 30441

This theorem is referenced by:  pliguhgr  30452