Theorem n0lplig 27678

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 27676	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		elirr 8656	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4386	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 220	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2778	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2839	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 318	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1629   ∈ wcel 2143  Vcvv 3348  ∅c0 4060  {csn 4313  Pligcplig 27669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033  ax-reg 8651

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-v 3350  df-dif 3723  df-un 3725  df-nul 4061  df-sn 4314  df-pr 4316  df-uni 4572  df-plig 27670

This theorem is referenced by:  pliguhgr  27681