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Theorem broutsideof 34816
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.)
Assertion
Ref Expression
broutsideof (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))

Proof of Theorem broutsideof
StepHypRef Expression
1 df-outsideof 34815 . . 3 OutsideOf = ( Colinear βˆ– Btwn )
21breqi 5131 . 2 (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ 𝑃( Colinear βˆ– Btwn )⟨𝐴, 𝐡⟩)
3 brdif 5178 . 2 (𝑃( Colinear βˆ– Btwn )⟨𝐴, 𝐡⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
42, 3bitri 274 1 (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 396   βˆ– cdif 3925  βŸ¨cop 4612   class class class wbr 5125   Btwn cbtwn 27935   Colinear ccolin 34732  OutsideOfcoutsideof 34814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3461  df-dif 3931  df-br 5126  df-outsideof 34815
This theorem is referenced by:  broutsideof2  34817  outsideofrflx  34822  outsidele  34827  outsideofcol  34828
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