Theorem tgbtwnne 25385

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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐺 ∈ TarskiG)
6		tgbtwntriv2.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 ∈ 𝑃)
8		tgbtwntriv2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ 𝑃)
10		tgbtwnne.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
11	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
12		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
13	12	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶))
14	11, 13	eleqtrrd 2704	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴))
15	1, 2, 3, 5, 7, 9, 14	axtgbtwnid 25365	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵)
16	15	eqcomd 2628	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐴)
17		tgbtwnne.2	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
18	17	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ≠ 𝐴)
19	18	neneqd 2799	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴)
20	16, 19	pm2.65da 600	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶)
21	20	neqned 2801	1 ⊢ (𝜑 → 𝐴 ≠ 𝐶)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgb 25348  df-trkg 25352

This theorem is referenced by:  mideulem2  25626  opphllem  25627  outpasch  25647  lnopp2hpgb  25655  lmieu  25676  dfcgra2  25721