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Theorem ncolne1 25420
 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 481 . . . 4 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 ncolne.y . . . . 5 (𝜑𝑌𝐵)
87adantr 481 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑌𝐵)
9 ncolne.z . . . . 5 (𝜑𝑍𝐵)
109adantr 481 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑍𝐵)
11 ncolne.x . . . . 5 (𝜑𝑋𝐵)
1211adantr 481 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋𝐵)
13 eqid 2621 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
142, 13, 4, 6, 12, 10tgbtwntriv1 25286 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15 simpr 477 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1615oveq1d 6619 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
1714, 16eleqtrd 2700 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
182, 3, 4, 6, 8, 10, 12, 17btwncolg1 25350 . . 3 ((𝜑𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
191, 18mtand 690 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
2019neqned 2797 1 (𝜑𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235  LineGclng 25236 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-trkgc 25247  df-trkgb 25248  df-trkgcb 25249  df-trkg 25252 This theorem is referenced by:  ncolne2  25421  tglineneq  25439  midexlem  25487  mideulem2  25526  outpasch  25547  hlpasch  25548  trgcopy  25596  trgcopyeulem  25597  acopy  25624  acopyeu  25625  cgrg3col4  25634  tgasa1  25639  isoas  25644
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