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Theorem tncp 24844
Description: There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)
Hypothesis
Ref Expression
tncp.1  |-  P  = 
U. L
Assertion
Ref Expression
tncp  |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  (
a  e.  l  /\  b  e.  l  /\  c  e.  l )
)
Distinct variable groups:    L, a,
b, c, l    P, a, b, c
Allowed substitution hint:    P( l)

Proof of Theorem tncp
StepHypRef Expression
1 tncp.1 . . . 4  |-  P  = 
U. L
21isplig 24843 . . 3  |-  ( L  e.  Plig  ->  ( L  e.  Plig  <->  ( A. a  e.  P  A. b  e.  P  ( a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  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.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) ) )
32ibi 241 . 2  |-  ( L  e.  Plig  ->  ( A. a  e.  P  A. b  e.  P  (
a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  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.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) )
43simp3d 1005 1  |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  (
a  e.  l  /\  b  e.  l  /\  c  e.  l )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   E!wreu 2811   U.cuni 4240   Pligcplig 24841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-reu 2816  df-v 3110  df-uni 4241  df-plig 24842
This theorem is referenced by:  lpni  24845
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