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Theorem tgbtwnne 28498
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 7447 . . . . . 6 ((𝜑𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
1411, 13eleqtrrd 2844 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
151, 2, 3, 5, 7, 9, 14axtgbtwnid 28474 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐵)
1615eqcomd 2743 . . 3 ((𝜑𝐴 = 𝐶) → 𝐵 = 𝐴)
17 tgbtwnne.2 . . . . 5 (𝜑𝐵𝐴)
1817adantr 480 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐵𝐴)
1918neneqd 2945 . . 3 ((𝜑𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
2016, 19pm2.65da 817 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
2120neqned 2947 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgb 28457  df-trkg 28461
This theorem is referenced by:  mideulem2  28742  opphllem  28743  outpasch  28763  lnopp2hpgb  28771  lmieu  28792  dfcgra2  28838
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