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Mirrors > Home > MPE Home > Th. List > Mathboxes > outsideofcol | Structured version Visualization version GIF version |
Description: Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
outsideofcol | ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 → 𝑃 Colinear 〈𝑄, 𝑅〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | broutsideof 33582 | . 2 ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 ↔ (𝑃 Colinear 〈𝑄, 𝑅〉 ∧ ¬ 𝑃 Btwn 〈𝑄, 𝑅〉)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 → 𝑃 Colinear 〈𝑄, 𝑅〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 〈cop 4573 class class class wbr 5066 Btwn cbtwn 26675 Colinear ccolin 33498 OutsideOfcoutsideof 33580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3939 df-br 5067 df-outsideof 33581 |
This theorem is referenced by: (None) |
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