Theorem tncp 27330

Description: In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.)

Hypothesis

Ref	Expression
tncp.1	|- P = U. G

Assertion

Ref	Expression
tncp	|- ( G e. Plig -> E. a e. P E. b e. P E. c e. P A. l e. G -. ( a e. l /\ b e. l /\ c e. l ) )

Distinct variable groups:   a, b, c, l, G   P, a, b, c

Allowed substitution hint:   P( l)

Proof of Theorem tncp

Step	Hyp	Ref	Expression
1		tncp.1	. . . 4 |- P = U. G
2	1	isplig 27328	. . 3 |- ( G e. Plig -> ( G e. Plig <-> ( A. a e. P A. b e. P ( a =/= b -> E! l e. G ( a e. l /\ b e. l ) ) /\ A. l e. G E. a e. P E. b e. P ( a =/= b /\ a e. l /\ b e. l ) /\ E. a e. P E. b e. P E. c e. P A. l e. G -. ( a e. l /\ b e. l /\ c e. l ) ) ) )
3	2	ibi 256	. 2 |- ( G e. Plig -> ( A. a e. P A. b e. P ( a =/= b -> E! l e. G ( a e. l /\ b e. l ) ) /\ A. l e. G E. a e. P E. b e. P ( a =/= b /\ a e. l /\ b e. l ) /\ E. a e. P E. b e. P E. c e. P A. l e. G -. ( a e. l /\ b e. l /\ c e. l ) ) )
4	3	simp3d 1075	1 |- ( G e. Plig -> E. a e. P E. b e. P E. c e. P A. l e. G -. ( a e. l /\ b e. l /\ c e. l ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   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-ral 2917  df-rex 2918  df-reu 2919  df-v 3202  df-uni 4437  df-plig 27327

This theorem is referenced by:  lpni  27332