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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwncomb.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnne.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnne.2	⊢ (𝜑 → 𝐵 ≠ 𝐴)

Assertion

Ref	Expression
tgbtwnne	⊢ (𝜑 → 𝐴 ≠ 𝐶)

Proof of Theorem tgbtwnne

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6		tgbtwntriv2.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
8		tgbtwntriv2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ 𝑃)
10		tgbtwnne.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
13	12	oveq2d 7430	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
14	11, 13	eleqtrrd 2831	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
15	1, 2, 3, 5, 7, 9, 14	axtgbtwnid 28244	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵)
16	15	eqcomd 2733	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐴)
17		tgbtwnne.2	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ≠ 𝐴)
19	18	neneqd 2940	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
20	16, 19	pm2.65da 816	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
21	20	neqned 2942	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

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