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Theorem broutsideof 30432
Description: Binary relationship 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 30431 . . 3  |- OutsideOf  =  ( 
Colinear  \  Btwn  )
21breqi 4400 . 2  |-  ( POutsideOf <. A ,  B >.  <->  P
(  Colinear  \  Btwn  ) <. A ,  B >. )
3 brdif 4444 . 2  |-  ( P (  Colinear  \  Btwn  ) <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
42, 3bitri 249 1  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    \ cdif 3410   <.cop 3977   class class class wbr 4394    Btwn cbtwn 24490    Colinear ccolin 30348  OutsideOfcoutsideof 30430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-dif 3416  df-br 4395  df-outsideof 30431
This theorem is referenced by:  broutsideof2  30433  outsideofrflx  30438  outsidele  30443  outsideofcol  30444
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