Theorem n0lplig 30411

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 30409	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5311	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4717	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 229	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2735	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2817	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 328	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3463  ∅c0 4323  {csn 4624  Pligcplig 30402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5295

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-v 3465  df-dif 3950  df-ss 3964  df-nul 4324  df-sn 4625  df-uni 4907  df-plig 30403

This theorem is referenced by:  pliguhgr  30414