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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
tglineelsb2.i Itv
tglineelsb2.l LineG
tglineelsb2.g TarskiG
ncolne.x
ncolne.y
ncolne.z
ncolne.2
Assertion
Ref Expression
ncolne2

Proof of Theorem ncolne2
StepHypRef Expression
1 tglineelsb2.p . 2
2 tglineelsb2.i . 2 Itv
3 tglineelsb2.l . 2 LineG
4 tglineelsb2.g . 2 TarskiG
5 ncolne.x . 2
6 ncolne.z . 2
7 ncolne.y . 2
8 ncolne.2 . . 3
91, 3, 2, 4, 7, 6, 5, 8ncolcom 23820 . 2
101, 2, 3, 4, 5, 6, 7, 9ncolne1 23877 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 368   wceq 1383   wcel 1804   wne 2638  cfv 5578  (class class class)co 6281  cbs 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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