Theorem n0lplig 31078

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 31076	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5274	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4678	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 233	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2770	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2852	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 332	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  {csn 4584  Pligcplig 31069

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-v 3453  df-dif 3902  df-ss 3916  df-nul 4280  df-sn 4585  df-uni 4868  df-plig 31070

This theorem is used by:  pliguhgr  31081