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Theorem ncolne1 28501
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 479 . . . 4 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 ncolne.y . . . . 5 (𝜑𝑌𝐵)
87adantr 479 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑌𝐵)
9 ncolne.z . . . . 5 (𝜑𝑍𝐵)
109adantr 479 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑍𝐵)
11 ncolne.x . . . . 5 (𝜑𝑋𝐵)
1211adantr 479 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋𝐵)
13 eqid 2725 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
142, 13, 4, 6, 12, 10tgbtwntriv1 28367 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15 simpr 483 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1615oveq1d 7434 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
1714, 16eleqtrd 2827 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
182, 3, 4, 6, 8, 10, 12, 17btwncolg1 28431 . . 3 ((𝜑𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
191, 18mtand 814 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
2019neqned 2936 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2929  cfv 6549  (class class class)co 7419  Basecbs 17183  distcds 17245  TarskiGcstrkg 28303  Itvcitv 28309  LineGclng 28310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-trkgc 28324  df-trkgb 28325  df-trkgcb 28326  df-trkg 28329
This theorem is referenced by:  ncolne2  28502  tglineneq  28520  midexlem  28568  mideulem2  28610  outpasch  28631  hlpasch  28632  trgcopy  28680  trgcopyeulem  28681  acopy  28709  acopyeu  28710  cgrg3col4  28729  tgasa1  28734
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