Theorem tgbtwnne 28009

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 7428	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
14	11, 13	eleqtrrd 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
15	1, 2, 3, 5, 7, 9, 14	axtgbtwnid 27985	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵)
16	15	eqcomd 2737	. . 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 2944	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
20	16, 19	pm2.65da 814	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
21	20	neqned 2946	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ‘cfv 6543  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27946  Itvcitv 27952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-trkgb 27968  df-trkg 27972

This theorem is referenced by:  mideulem2  28253  opphllem  28254  outpasch  28274  lnopp2hpgb  28282  lmieu  28303  dfcgra2  28349