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Theorem n0lplig 28245
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 28243 . 2 (𝐺 ∈ Plig → ¬ {V} ∈ 𝐺)
2 vprc 5192 . . . . 5 ¬ V ∈ V
3 snprc 4626 . . . . 5 (¬ V ∈ V ↔ {V} = ∅)
42, 3mpbi 233 . . . 4 {V} = ∅
54eqcomi 2830 . . 3 ∅ = {V}
65eleq1i 2902 . 2 (∅ ∈ 𝐺 ↔ {V} ∈ 𝐺)
71, 6sylnibr 332 1 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2115  Vcvv 3471  c0 4266  {csn 4540  Pligcplig 28236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-v 3473  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4267  df-sn 4541  df-uni 4812  df-plig 28237
This theorem is referenced by:  pliguhgr  28248
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