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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 7430 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶)) |
14 | 11, 13 | eleqtrrd 2831 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 28244 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵) |
16 | 15 | eqcomd 2733 | . . 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 2940 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴) |
20 | 16, 19 | pm2.65da 816 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶) |
21 | 20 | neqned 2942 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 distcds 17227 TarskiGcstrkg 28205 Itvcitv 28211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-trkgb 28227 df-trkg 28231 |
This theorem is referenced by: mideulem2 28512 opphllem 28513 outpasch 28533 lnopp2hpgb 28541 lmieu 28562 dfcgra2 28608 |
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