Theorem n0lplig 30455

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 30453	. 2 ⊢ (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2		vprc 5248	. . . . 5 ⊢ ¬ V ∈ V
3		snprc 4665	. . . . 5 ⊢ (¬ V ∈ V ↔ {V} = ∅)
4	2, 3	mpbi 230	. . . 4 ⊢ {V} = ∅
5	4	eqcomi 2740	. . 3 ⊢ ∅ = {V}
6	5	eleq1i 2822	. 2 ⊢ (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
7	1, 6	sylnibr 329	1 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4278  {csn 4571  Pligcplig 30446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-v 3438  df-dif 3900  df-ss 3914  df-nul 4279  df-sn 4572  df-uni 4855  df-plig 30447

This theorem is referenced by:  pliguhgr  30458