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Theorem ncolne1 26425
 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 484 . . . 4 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 ncolne.y . . . . 5 (𝜑𝑌𝐵)
87adantr 484 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑌𝐵)
9 ncolne.z . . . . 5 (𝜑𝑍𝐵)
109adantr 484 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑍𝐵)
11 ncolne.x . . . . 5 (𝜑𝑋𝐵)
1211adantr 484 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋𝐵)
13 eqid 2824 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
142, 13, 4, 6, 12, 10tgbtwntriv1 26291 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15 simpr 488 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1615oveq1d 7164 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
1714, 16eleqtrd 2918 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
182, 3, 4, 6, 8, 10, 12, 17btwncolg1 26355 . . 3 ((𝜑𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
191, 18mtand 815 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
2019neqned 3021 1 (𝜑𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  distcds 16574  TarskiGcstrkg 26230  Itvcitv 26236  LineGclng 26237 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-trkgc 26248  df-trkgb 26249  df-trkgcb 26250  df-trkg 26253 This theorem is referenced by:  ncolne2  26426  tglineneq  26444  midexlem  26492  mideulem2  26534  outpasch  26555  hlpasch  26556  trgcopy  26604  trgcopyeulem  26605  acopy  26633  acopyeu  26634  cgrg3col4  26653  tgasa1  26658
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