MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwnne Structured version   Visualization version   GIF version

Theorem tgbtwnne 26581
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 484 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . . . 6 (𝜑𝐴𝑃)
76adantr 484 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . . . 6 (𝜑𝐵𝑃)
98adantr 484 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵𝑃)
10 tgbtwnne.1 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1110adantr 484 . . . . . 6 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12 simpr 488 . . . . . . 7 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7229 . . . . . 6 ((𝜑𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
1411, 13eleqtrrd 2841 . . . . 5 ((𝜑𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
151, 2, 3, 5, 7, 9, 14axtgbtwnid 26557 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐵)
1615eqcomd 2743 . . 3 ((𝜑𝐴 = 𝐶) → 𝐵 = 𝐴)
17 tgbtwnne.2 . . . . 5 (𝜑𝐵𝐴)
1817adantr 484 . . . 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 399   = wceq 1543  wcel 2110  wne 2940  cfv 6380  (class class class)co 7213  Basecbs 16760  distcds 16811  TarskiGcstrkg 26521  Itvcitv 26527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-trkgb 26540  df-trkg 26544
This theorem is referenced by:  mideulem2  26825  opphllem  26826  outpasch  26846  lnopp2hpgb  26854  lmieu  26875  dfcgra2  26921
  Copyright terms: Public domain W3C validator