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Theorem broutsideof 33731
 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, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))

Proof of Theorem broutsideof
StepHypRef Expression
1 df-outsideof 33730 . . 3 OutsideOf = ( Colinear ∖ Btwn )
21breqi 5037 . 2 (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃( Colinear ∖ Btwn )⟨𝐴, 𝐵⟩)
3 brdif 5084 . 2 (𝑃( Colinear ∖ Btwn )⟨𝐴, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
42, 3bitri 278 1 (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∖ cdif 3878  ⟨cop 4531   class class class wbr 5031   Btwn cbtwn 26697   Colinear ccolin 33647  OutsideOfcoutsideof 33729 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-br 5032  df-outsideof 33730 This theorem is referenced by:  broutsideof2  33732  outsideofrflx  33737  outsidele  33742  outsideofcol  33743
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