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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 1252 | . 2 ⊢ (𝐶 Btwn 〈𝐴, 𝐵〉 → (𝐴 Btwn 〈𝐵, 𝐶〉 ∨ 𝐵 Btwn 〈𝐶, 𝐴〉 ∨ 𝐶 Btwn 〈𝐴, 𝐵〉)) | |
2 | brcolinear 32291 | . 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 236 | 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 383 ∨ w3o 1053 ∧ w3a 1054 ∈ wcel 2030 〈cop 4216 class class class wbr 4685 ‘cfv 5926 ℕcn 11058 𝔼cee 25813 Btwn cbtwn 25814 Colinear ccolin 32269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 df-cnv 5151 df-iota 5889 df-fv 5934 df-oprab 6694 df-colinear 32271 |
This theorem is referenced by: btwncolinear2 32302 btwncolinear3 32303 btwncolinear4 32304 btwncolinear5 32305 btwncolinear6 32306 idinside 32316 btwnconn1lem12 32330 brsegle2 32341 broutsideof2 32354 outsidele 32364 |
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