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Theorem broutsideof 24744
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 24743 . . 3  |- OutsideOf  =  ( 
Colinear  \  Btwn  )
21breqi 4029 . 2  |-  ( POutsideOf <. A ,  B >.  <->  P
(  Colinear  \  Btwn  ) <. A ,  B >. )
3 brdif 4071 . 2  |-  ( P (  Colinear  \  Btwn  ) <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
42, 3bitri 240 1  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    \ cdif 3149   <.cop 3643   class class class wbr 4023    Btwn cbtwn 24517    Colinear ccolin 24660  OutsideOfcoutsideof 24742
This theorem is referenced by:  broutsideof2  24745  outsideofrflx  24750  outsidele  24755  outsideofcol  24756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-br 4024  df-outsideof 24743
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