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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 1328 | . 2 ⊢ (𝐶 Btwn 〈𝐴, 𝐵〉 → (𝐴 Btwn 〈𝐵, 𝐶〉 ∨ 𝐵 Btwn 〈𝐶, 𝐴〉 ∨ 𝐶 Btwn 〈𝐴, 𝐵〉)) | |
2 | brcolinear 33515 | . 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 | syl5ibr 248 | 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 398 ∨ w3o 1082 ∧ w3a 1083 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ‘cfv 6349 ℕcn 11632 𝔼cee 26668 Btwn cbtwn 26669 Colinear ccolin 33493 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-iota 6308 df-fv 6357 df-oprab 7154 df-colinear 33495 |
This theorem is referenced by: btwncolinear2 33526 btwncolinear3 33527 btwncolinear4 33528 btwncolinear5 33529 btwncolinear6 33530 idinside 33540 btwnconn1lem12 33554 brsegle2 33565 broutsideof2 33578 outsidele 33588 |
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