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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 33103 | . 2 ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 ↔ (𝑃 Colinear 〈𝑄, 𝑅〉 ∧ ¬ 𝑃 Btwn 〈𝑄, 𝑅〉)) | |
2 | 1 | simplbi 490 | 1 ⊢ (𝑃OutsideOf〈𝑄, 𝑅〉 → 𝑃 Colinear 〈𝑄, 𝑅〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 〈cop 4441 class class class wbr 4923 Btwn cbtwn 26372 Colinear ccolin 33019 OutsideOfcoutsideof 33101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-dif 3826 df-br 4924 df-outsideof 33102 |
This theorem is referenced by: (None) |
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