Theorem broutsideof 36163

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 36162	. . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn )
2	1	breqi 5095	. 2 ⊢ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃( Colinear ∖ Btwn )⟨𝐴, 𝐵⟩)
3		brdif 5142	. 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 206   ∧ wa 395   ∖ cdif 3894  ⟨cop 4579   class class class wbr 5089   Btwn cbtwn 28867   Colinear ccolin 36079  OutsideOfcoutsideof 36161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-br 5090  df-outsideof 36162

This theorem is referenced by:  broutsideof2  36164  outsideofrflx  36169  outsidele  36174  outsideofcol  36175