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Theorem tgbtwnne 26849
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 481 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . . . 6 (𝜑𝐴𝑃)
76adantr 481 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . . . 6 (𝜑𝐵𝑃)
98adantr 481 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵𝑃)
10 tgbtwnne.1 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1110adantr 481 . . . . . 6 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12 simpr 485 . . . . . . 7 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7287 . . . . . 6 ((𝜑𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
1411, 13eleqtrrd 2844 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
151, 2, 3, 5, 7, 9, 14axtgbtwnid 26825 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐵)
1615eqcomd 2746 . . 3 ((𝜑𝐴 = 𝐶) → 𝐵 = 𝐴)
17 tgbtwnne.2 . . . . 5 (𝜑𝐵𝐴)
1817adantr 481 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐵𝐴)
1918neneqd 2950 . . 3 ((𝜑𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
2016, 19pm2.65da 814 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
2120neqned 2952 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  wne 2945  cfv 6432  (class class class)co 7271  Basecbs 16910  distcds 16969  TarskiGcstrkg 26786  Itvcitv 26792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-nul 5234
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6390  df-fv 6440  df-ov 7274  df-trkgb 26808  df-trkg 26812
This theorem is referenced by:  mideulem2  27093  opphllem  27094  outpasch  27114  lnopp2hpgb  27122  lmieu  27143  dfcgra2  27189
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