Theorem broutsideof 32228

Description: Binary relation 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

Step	Hyp	Ref	Expression
1		df-outsideof 32227	. . 3 |- OutsideOf = ( Colinear \ Btwn )
2	1	breqi 4659	. 2 |- ( POutsideOf <. A , B >. <-> P ( Colinear \ Btwn ) <. A , B >. )
3		brdif 4705	. 2 |- ( P ( Colinear \ Btwn ) <. A , B >. <-> ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) )
4	2, 3	bitri 264	1 |- ( POutsideOf <. A , B >. <-> ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   \ cdif 3571   <.cop 4183   class class class wbr 4653   Btwn cbtwn 25769   Colinear ccolin 32144  OutsideOfcoutsideof 32226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-br 4654  df-outsideof 32227

This theorem is referenced by:  broutsideof2  32229  outsideofrflx  32234  outsidele  32239  outsideofcol  32240