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| Mirrors > Home > MPE Home > Th. List > Mathboxes > broutsideof | Structured version Visualization version GIF version | ||
| Description: Binary relation form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| Ref | Expression |
|---|---|
| broutsideof | ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-outsideof 36121 | . . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn ) | |
| 2 | 1 | breqi 5149 | . 2 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ 𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉) |
| 3 | brdif 5196 | . 2 ⊢ (𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∖ cdif 3948 〈cop 4632 class class class wbr 5143 Btwn cbtwn 28904 Colinear ccolin 36038 OutsideOfcoutsideof 36120 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-br 5144 df-outsideof 36121 |
| This theorem is referenced by: broutsideof2 36123 outsideofrflx 36128 outsidele 36133 outsideofcol 36134 |
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