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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 7447 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → (𝐴𝐼𝐴) = (𝐴𝐼𝐶)) | 
| 14 | 11, 13 | eleqtrrd 2844 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐴)) | 
| 15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 28474 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐵) | 
| 16 | 15 | eqcomd 2743 | . . 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 2945 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → ¬ 𝐵 = 𝐴) | 
| 20 | 16, 19 | pm2.65da 817 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐶) | 
| 21 | 20 | neqned 2947 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-trkgb 28457 df-trkg 28461 | 
| This theorem is referenced by: mideulem2 28742 opphllem 28743 outpasch 28763 lnopp2hpgb 28771 lmieu 28792 dfcgra2 28838 | 
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