Theorem n0lplig 27335

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 e. Plig -> -. (/) e. G )

Proof of Theorem n0lplig

Step	Hyp	Ref	Expression
1		nsnlplig 27333	. 2 |- ( G e. Plig -> -. { _V } e. G )
2		elirr 8505	. . . . 5 |- -. _V e. _V
3		snprc 4253	. . . . 5 |- ( -. _V e. _V <-> { _V } = (/) )
4	2, 3	mpbi 220	. . . 4 |- { _V } = (/)
5	4	eqcomi 2631	. . 3 |- (/) = { _V }
6	5	eleq1i 2692	. 2 |- ( (/) e. G <-> { _V } e. G )
7	1, 6	sylnibr 319	1 |- ( G e. Plig -> -. (/) e. G )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   Pligcplig 27326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437  df-plig 27327

This theorem is referenced by:  pliguhgr  27338