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Theorem broutsideof 24152
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 24151 . . 3  |- OutsideOf  =  ( 
Colinear  \  Btwn  )
21breqi 4031 . 2  |-  ( POutsideOf <. A ,  B >.  <->  P
(  Colinear  \  Btwn  ) <. A ,  B >. )
3 brdif 4073 . 2  |-  ( P (  Colinear  \  Btwn  ) <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
42, 3bitri 242 1  |-  ( POutsideOf <. A ,  B >.  <->  ( P  Colinear  <. A ,  B >.  /\  -.  P  Btwn  <. A ,  B >. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360    \ cdif 3151   <.cop 3645   class class class wbr 4025    Btwn cbtwn 23925    Colinear ccolin 24068  OutsideOfcoutsideof 24150
This theorem is referenced by:  broutsideof2  24153  outsideofrflx  24158  outsidele  24163  outsideofcol  24164
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-dif 3157  df-br 4026  df-outsideof 24151
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