Theorem nsnlplig 27333

Description: There is no "one-point line" in a planar incidence geometry. (Contributed by BJ, 2-Dec-2021.) (Proof shortened by AV, 5-Dec-2021.)

Assertion

Ref	Expression
nsnlplig	|- ( G e. Plig -> -. { A } e. G )

Proof of Theorem nsnlplig

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- U. G = U. G
2	1	l2p 27331	. . 3 |- ( ( G e. Plig /\ { A } e. G ) -> E. a e. U. G E. b e. U. G ( a =/= b /\ a e. { A } /\ b e. { A } ) )
3		elsni 4194	. . . . . . . 8 |- ( a e. { A } -> a = A )
4		elsni 4194	. . . . . . . 8 |- ( b e. { A } -> b = A )
5		eqtr3 2643	. . . . . . . . 9 |- ( ( a = A /\ b = A ) -> a = b )
6		eqneqall 2805	. . . . . . . . 9 |- ( a = b -> ( a =/= b -> -. { A } e. G ) )
7	5, 6	syl 17	. . . . . . . 8 |- ( ( a = A /\ b = A ) -> ( a =/= b -> -. { A } e. G ) )
8	3, 4, 7	syl2an 494	. . . . . . 7 |- ( ( a e. { A } /\ b e. { A } ) -> ( a =/= b -> -. { A } e. G ) )
9	8	impcom 446	. . . . . 6 |- ( ( a =/= b /\ ( a e. { A } /\ b e. { A } ) ) -> -. { A } e. G )
10	9	3impb 1260	. . . . 5 |- ( ( a =/= b /\ a e. { A } /\ b e. { A } ) -> -. { A } e. G )
11	10	a1i 11	. . . 4 |- ( ( a e. U. G /\ b e. U. G ) -> ( ( a =/= b /\ a e. { A } /\ b e. { A } ) -> -. { A } e. G ) )
12	11	rexlimivv 3036	. . 3 |- ( E. a e. U. G E. b e. U. G ( a =/= b /\ a e. { A } /\ b e. { A } ) -> -. { A } e. G )
13	2, 12	syl 17	. 2 |- ( ( G e. Plig /\ { A } e. G ) -> -. { A } e. G )
14	13	pm2.01da 458	1 |- ( G e. Plig -> -. { A } e. G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {csn 4177   U.cuni 4436   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

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-sn 4178  df-uni 4437  df-plig 27327

This theorem is referenced by:  n0lplig  27335