![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tgbtwnne | Structured version Visualization version GIF version |
Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
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) |
Ref | Expression |
---|---|
tgbtwnne | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
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 7447 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶)) |
14 | 11, 13 | eleqtrrd 2842 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 28489 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵) |
16 | 15 | eqcomd 2741 | . . 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 2943 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴) |
20 | 16, 19 | pm2.65da 817 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶) |
21 | 20 | neqned 2945 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-trkgb 28472 df-trkg 28476 |
This theorem is referenced by: mideulem2 28757 opphllem 28758 outpasch 28778 lnopp2hpgb 28786 lmieu 28807 dfcgra2 28853 |
Copyright terms: Public domain | W3C validator |