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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 36062 | . . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn ) | |
2 | 1 | breqi 5156 | . 2 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ 𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉) |
3 | brdif 5203 | . 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 3960 〈cop 4637 class class class wbr 5150 Btwn cbtwn 28901 Colinear ccolin 35979 OutsideOfcoutsideof 36061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1538 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-v 3479 df-dif 3966 df-br 5151 df-outsideof 36062 |
This theorem is referenced by: broutsideof2 36064 outsideofrflx 36069 outsidele 36074 outsideofcol 36075 |
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