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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwncolinear1 | Structured version Visualization version GIF version |
Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.) |
Ref | Expression |
---|---|
btwncolinear1 | ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐴 Colinear 〈𝐵, 𝐶〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3mix3 1331 | . 2 ⊢ (𝐶 Btwn 〈𝐴, 𝐵〉 → (𝐴 Btwn 〈𝐵, 𝐶〉 ∨ 𝐵 Btwn 〈𝐶, 𝐴〉 ∨ 𝐶 Btwn 〈𝐴, 𝐵〉)) | |
2 | brcolinear 36040 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear 〈𝐵, 𝐶〉 ↔ (𝐴 Btwn 〈𝐵, 𝐶〉 ∨ 𝐵 Btwn 〈𝐶, 𝐴〉 ∨ 𝐶 Btwn 〈𝐴, 𝐵〉))) | |
3 | 1, 2 | imbitrrid 246 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn 〈𝐴, 𝐵〉 → 𝐴 Colinear 〈𝐵, 𝐶〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 ∈ wcel 2105 〈cop 4636 class class class wbr 5147 ‘cfv 6562 ℕcn 12263 𝔼cee 28917 Btwn cbtwn 28918 Colinear ccolin 36018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-iota 6515 df-fv 6570 df-oprab 7434 df-colinear 36020 |
This theorem is referenced by: btwncolinear2 36051 btwncolinear3 36052 btwncolinear4 36053 btwncolinear5 36054 btwncolinear6 36055 idinside 36065 btwnconn1lem12 36079 brsegle2 36090 broutsideof2 36103 outsidele 36113 |
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