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Theorem tgbtwnne 25606
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 466 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . . . 6 (𝜑𝐴𝑃)
76adantr 466 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . . . 6 (𝜑𝐵𝑃)
98adantr 466 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵𝑃)
10 tgbtwnne.1 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1110adantr 466 . . . . . 6 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12 simpr 471 . . . . . . 7 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 6809 . . . . . 6 ((𝜑𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
1411, 13eleqtrrd 2853 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
151, 2, 3, 5, 7, 9, 14axtgbtwnid 25586 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐵)
1615eqcomd 2777 . . 3 ((𝜑𝐴 = 𝐶) → 𝐵 = 𝐴)
17 tgbtwnne.2 . . . . 5 (𝜑𝐵𝐴)
1817adantr 466 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐵𝐴)
1918neneqd 2948 . . 3 ((𝜑𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
2016, 19pm2.65da 818 . 2 (𝜑 → ¬ 𝐴 = 𝐶)
2120neqned 2950 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgb 25569  df-trkg 25573
This theorem is referenced by:  mideulem2  25847  opphllem  25848  outpasch  25868  lnopp2hpgb  25876  lmieu  25897  dfcgra2  25942
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