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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 34361 | . 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 245 | 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 396 ∨ w3o 1085 ∧ w3a 1086 ∈ wcel 2106 〈cop 4567 class class class wbr 5074 ‘cfv 6433 ℕcn 11973 𝔼cee 27256 Btwn cbtwn 27257 Colinear ccolin 34339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-iota 6391 df-fv 6441 df-oprab 7279 df-colinear 34341 |
This theorem is referenced by: btwncolinear2 34372 btwncolinear3 34373 btwncolinear4 34374 btwncolinear5 34375 btwncolinear6 34376 idinside 34386 btwnconn1lem12 34400 brsegle2 34411 broutsideof2 34424 outsidele 34434 |
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