![]() |
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 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) |
Colors of variables: wff setvar class |
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 |
Copyright terms: Public domain | W3C validator |