Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  broutsideof Structured version   Visualization version   GIF version

Theorem broutsideof 35397
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 35396 . . 3 OutsideOf = ( Colinear βˆ– Btwn )
21breqi 5153 . 2 (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ 𝑃( Colinear βˆ– Btwn )⟨𝐴, 𝐡⟩)
3 brdif 5200 . 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 394   βˆ– cdif 3944  βŸ¨cop 4633   class class class wbr 5147   Btwn cbtwn 28414   Colinear ccolin 35313  OutsideOfcoutsideof 35395
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-br 5148  df-outsideof 35396
This theorem is referenced by:  broutsideof2  35398  outsideofrflx  35403  outsidele  35408  outsideofcol  35409
  Copyright terms: Public domain W3C validator