Theorem broutsideof 36116

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 36115	. . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn )
2	1	breqi 5116	. 2 ⊢ (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃( Colinear ∖ Btwn )⟨𝐴, 𝐵⟩)
3		brdif 5163	. 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 3914  ⟨cop 4598   class class class wbr 5110   Btwn cbtwn 28823   Colinear ccolin 36032  OutsideOfcoutsideof 36114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-br 5111  df-outsideof 36115

This theorem is referenced by:  broutsideof2  36117  outsideofrflx  36122  outsidele  36127  outsideofcol  36128