MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0lplig Structured version   Visualization version   GIF version

Theorem n0lplig 27667
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
StepHypRef Expression
1 nsnlplig 27665 . 2 (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2 elirr 8669 . . . . 5 ¬ V ∈ V
3 snprc 4397 . . . . 5 (¬ V ∈ V ↔ {V} = ∅)
42, 3mpbi 220 . . . 4 {V} = ∅
54eqcomi 2769 . . 3 ∅ = {V}
65eleq1i 2830 . 2 (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
71, 6sylnibr 318 1 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  c0 4058  {csn 4321  Pligcplig 27658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-reg 8664
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324  df-uni 4589  df-plig 27659
This theorem is referenced by:  pliguhgr  27670
  Copyright terms: Public domain W3C validator