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

Theorem broutsideof 24084
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 24083 . . 3  |- OutsideOf  =  ( 
Colinear  \  Btwn  )
21breqi 3969 . 2  |-  ( POutsideOf <. A ,  B >.  <->  P
(  Colinear  \  Btwn  ) <. A ,  B >. )
3 brdif 4011 . 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 3091   <.cop 3584   class class class wbr 3963    Btwn cbtwn 23857    Colinear ccolin 24000  OutsideOfcoutsideof 24082
This theorem is referenced by:  broutsideof2  24085  outsideofrflx  24090  outsidele  24095  outsideofcol  24096
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-dif 3097  df-br 3964  df-outsideof 24083
  Copyright terms: Public domain W3C validator