Theorem tgbtwnne 25993

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 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6		tgbtwntriv2.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
8		tgbtwntriv2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ 𝑃)
10		tgbtwnne.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
11	10	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
13	12	oveq2d 6990	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
14	11, 13	eleqtrrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
15	1, 2, 3, 5, 7, 9, 14	axtgbtwnid 25969	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵)
16	15	eqcomd 2777	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐴)
17		tgbtwnne.2	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
18	17	adantr 473	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ≠ 𝐴)
19	18	neneqd 2965	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
20	16, 19	pm2.65da 805	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
21	20	neqned 2967	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051   ≠ wne 2960  ‘cfv 6185  (class class class)co 6974  Basecbs 16337  distcds 16428  TarskiGcstrkg 25933  Itvcitv 25939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-nul 5063

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-iota 6149  df-fv 6193  df-ov 6977  df-trkgb 25952  df-trkg 25956

This theorem is referenced by:  mideulem2  26237  opphllem  26238  outpasch  26258  lnopp2hpgb  26266  lmieu  26287  dfcgra2  26333