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

Theorem broutsideof 24816
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 24815 . . 3  |- OutsideOf  =  ( 
Colinear  \  Btwn  )
21breqi 4045 . 2  |-  ( POutsideOf <. A ,  B >.  <->  P
(  Colinear  \  Btwn  ) <. A ,  B >. )
3 brdif 4087 . 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 3162   <.cop 3656   class class class wbr 4039    Btwn cbtwn 24589    Colinear ccolin 24732  OutsideOfcoutsideof 24814
This theorem is referenced by:  broutsideof2  24817  outsideofrflx  24822  outsidele  24827  outsideofcol  24828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-br 4040  df-outsideof 24815
  Copyright terms: Public domain W3C validator