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Theorem ncolne2 23878
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 23878 could be simplified out and deleted, replaced by ncolcom 23820.
Hypotheses
Ref Expression
tglineelsb2.p  |-  B  =  ( Base `  G
)
tglineelsb2.i  |-  I  =  (Itv `  G )
tglineelsb2.l  |-  L  =  (LineG `  G )
tglineelsb2.g  |-  ( ph  ->  G  e. TarskiG )
ncolne.x  |-  ( ph  ->  X  e.  B )
ncolne.y  |-  ( ph  ->  Y  e.  B )
ncolne.z  |-  ( ph  ->  Z  e.  B )
ncolne.2  |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
Assertion
Ref Expression
ncolne2  |-  ( ph  ->  X  =/=  Z )

Proof of Theorem ncolne2
StepHypRef Expression
1 tglineelsb2.p . 2  |-  B  =  ( Base `  G
)
2 tglineelsb2.i . 2  |-  I  =  (Itv `  G )
3 tglineelsb2.l . 2  |-  L  =  (LineG `  G )
4 tglineelsb2.g . 2  |-  ( ph  ->  G  e. TarskiG )
5 ncolne.x . 2  |-  ( ph  ->  X  e.  B )
6 ncolne.z . 2  |-  ( ph  ->  Z  e.  B )
7 ncolne.y . 2  |-  ( ph  ->  Y  e.  B )
8 ncolne.2 . . 3  |-  ( ph  ->  -.  ( X  e.  ( Y L Z )  \/  Y  =  Z ) )
91, 3, 2, 4, 7, 6, 5, 8ncolcom 23820 . 2  |-  ( ph  ->  -.  ( X  e.  ( Z L Y )  \/  Z  =  Y ) )
101, 2, 3, 4, 5, 6, 7, 9ncolne1 23877 1  |-  ( ph  ->  X  =/=  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   Basecbs 14509  TarskiGcstrkg 23697  Itvcitv 23704  LineGclng 23705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-trkgc 23716  df-trkgb 23717  df-trkgcb 23718  df-trkg 23722
This theorem is referenced by:  midexlem  23941
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