Theorem n0lplig 31142

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 31140	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5281	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4680	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 233	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2772	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2855	. 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 2147  Vcvv 3454  ∅c0 4281  {csn 4588  Pligcplig 31133

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282  df-sn 4589  df-uni 4873  df-plig 31134

This theorem is used by:  pliguhgr  31145