Theorem broutsideof 35093

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

Step	Hyp	Ref	Expression
1		df-outsideof 35092	. . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn )
2	1	breqi 5155	. 2 ⊢ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃( Colinear ∖ Btwn )⟨𝐴, 𝐵⟩)
3		brdif 5202	. 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⟩))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∖ cdif 3946  ⟨cop 4635   class class class wbr 5149   Btwn cbtwn 28147   Colinear ccolin 35009  OutsideOfcoutsideof 35091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-br 5150  df-outsideof 35092

This theorem is referenced by:  broutsideof2  35094  outsideofrflx  35099  outsidele  35104  outsideofcol  35105