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Theorem ncolne2 26424
 Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) TODO (NM): maybe ncolne2 26424 could be simplified out and deleted, replaced by ncolcom 26359.
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
ncolne.x (𝜑𝑋𝐵)
ncolne.y (𝜑𝑌𝐵)
ncolne.z (𝜑𝑍𝐵)
ncolne.2 (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
Assertion
Ref Expression
ncolne2 (𝜑𝑋𝑍)

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 (𝜑𝐺 ∈ TarskiG)
5 ncolne.x . 2 (𝜑𝑋𝐵)
6 ncolne.z . 2 (𝜑𝑍𝐵)
7 ncolne.y . 2 (𝜑𝑌𝐵)
8 ncolne.2 . . 3 (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
91, 3, 2, 4, 7, 6, 5, 8ncolcom 26359 . 2 (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
101, 2, 3, 4, 5, 6, 7, 9ncolne1 26423 1 (𝜑𝑋𝑍)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  TarskiGcstrkg 26228  Itvcitv 26234  LineGclng 26235 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-trkgc 26246  df-trkgb 26247  df-trkgcb 26248  df-trkg 26251 This theorem is referenced by:  midexlem  26490
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