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Theorem tgbtwnne 28268
Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
tgbtwncomb.3 (𝜑𝐶𝑃)
tgbtwnne.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnne.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
tgbtwnne (𝜑𝐴𝐶)

Proof of Theorem tgbtwnne
StepHypRef Expression
1 tkgeom.p . . . . 5 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . . 5 = (dist‘𝐺)
3 tkgeom.i . . . . 5 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . . . 6 (𝜑𝐴𝑃)
76adantr 480 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . . . 6 (𝜑𝐵𝑃)
98adantr 480 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵𝑃)
10 tgbtwnne.1 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1110adantr 480 . . . . . 6 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12 simpr 484 . . . . . . 7 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7430 . . . . . 6 ((𝜑𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
1411, 13eleqtrrd 2831 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
151, 2, 3, 5, 7, 9, 14axtgbtwnid 28244 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐵)
1615eqcomd 2733 . . 3 ((𝜑𝐴 = 𝐶) → 𝐵 = 𝐴)
17 tgbtwnne.2 . . . . 5 (𝜑𝐵𝐴)
1817adantr 480 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐵𝐴)
1918neneqd 2940 . . 3 ((𝜑𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
2016, 19pm2.65da 816 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
2120neqned 2942 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2935  cfv 6542  (class class class)co 7414  Basecbs 17165  distcds 17227  TarskiGcstrkg 28205  Itvcitv 28211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-trkgb 28227  df-trkg 28231
This theorem is referenced by:  mideulem2  28512  opphllem  28513  outpasch  28533  lnopp2hpgb  28541  lmieu  28562  dfcgra2  28608
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