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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 34117 | . 2 ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 ↔ (𝑃 Colinear 〈𝑄, 𝑅〉 ∧ ¬ 𝑃 Btwn 〈𝑄, 𝑅〉)) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 → 𝑃 Colinear 〈𝑄, 𝑅〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 〈cop 4537 class class class wbr 5043 Btwn cbtwn 26952 Colinear ccolin 34033 OutsideOfcoutsideof 34115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-v 3403 df-dif 3860 df-br 5044 df-outsideof 34116 |
This theorem is referenced by: (None) |
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