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Theorem ncolne1 25976
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)
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
ncolne1 (𝜑𝑋𝑌)

Proof of Theorem ncolne1
StepHypRef Expression
1 ncolne.2 . . 3 (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
2 tglineelsb2.p . . . 4 𝐵 = (Base‘𝐺)
3 tglineelsb2.l . . . 4 𝐿 = (LineG‘𝐺)
4 tglineelsb2.i . . . 4 𝐼 = (Itv‘𝐺)
5 tglineelsb2.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
65adantr 474 . . . 4 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 ncolne.y . . . . 5 (𝜑𝑌𝐵)
87adantr 474 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑌𝐵)
9 ncolne.z . . . . 5 (𝜑𝑍𝐵)
109adantr 474 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑍𝐵)
11 ncolne.x . . . . 5 (𝜑𝑋𝐵)
1211adantr 474 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋𝐵)
13 eqid 2778 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
142, 13, 4, 6, 12, 10tgbtwntriv1 25842 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15 simpr 479 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1615oveq1d 6937 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
1714, 16eleqtrd 2861 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
182, 3, 4, 6, 8, 10, 12, 17btwncolg1 25906 . . 3 ((𝜑𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
191, 18mtand 806 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
2019neqned 2976 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 836   = wceq 1601  wcel 2107  wne 2969  cfv 6135  (class class class)co 6922  Basecbs 16255  distcds 16347  TarskiGcstrkg 25781  Itvcitv 25787  LineGclng 25788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-trkgc 25799  df-trkgb 25800  df-trkgcb 25801  df-trkg 25804
This theorem is referenced by:  ncolne2  25977  tglineneq  25995  midexlem  26043  mideulem2  26082  outpasch  26103  hlpasch  26104  trgcopy  26152  trgcopyeulem  26153  acopy  26182  acopyeu  26183  cgrg3col4  26202  tgasa1  26207  isoas  26213
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