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Theorem ncolne1 25565
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 480 . . . 4 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 ncolne.y . . . . 5 (𝜑𝑌𝐵)
87adantr 480 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑌𝐵)
9 ncolne.z . . . . 5 (𝜑𝑍𝐵)
109adantr 480 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑍𝐵)
11 ncolne.x . . . . 5 (𝜑𝑋𝐵)
1211adantr 480 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋𝐵)
13 eqid 2651 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
142, 13, 4, 6, 12, 10tgbtwntriv1 25431 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍))
15 simpr 476 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1615oveq1d 6705 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍))
1714, 16eleqtrd 2732 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍))
182, 3, 4, 6, 8, 10, 12, 17btwncolg1 25495 . . 3 ((𝜑𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
191, 18mtand 692 . 2 (𝜑 → ¬ 𝑋 = 𝑌)
2019neqned 2830 1 (𝜑𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  cfv 5926  (class class class)co 6690  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkg 25397
This theorem is referenced by:  ncolne2  25566  tglineneq  25584  midexlem  25632  mideulem2  25671  outpasch  25692  hlpasch  25693  trgcopy  25741  trgcopyeulem  25742  acopy  25769  acopyeu  25770  cgrg3col4  25779  tgasa1  25784  isoas  25789
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