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Mirrors > Home > MPE Home > Th. List > Mathboxes > broutsideof | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
broutsideof | β’ (πOutsideOfβ¨π΄, π΅β© β (π Colinear β¨π΄, π΅β© β§ Β¬ π Btwn β¨π΄, π΅β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-outsideof 34815 | . . 3 β’ OutsideOf = ( Colinear β Btwn ) | |
2 | 1 | breqi 5131 | . 2 β’ (πOutsideOfβ¨π΄, π΅β© β π( Colinear β Btwn )β¨π΄, π΅β©) |
3 | brdif 5178 | . 2 β’ (π( Colinear β Btwn )β¨π΄, π΅β© β (π Colinear β¨π΄, π΅β© β§ Β¬ π Btwn β¨π΄, π΅β©)) | |
4 | 2, 3 | bitri 274 | 1 β’ (πOutsideOfβ¨π΄, π΅β© β (π Colinear β¨π΄, π΅β© β§ Β¬ π Btwn β¨π΄, π΅β©)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 β§ wa 396 β cdif 3925 β¨cop 4612 class class class wbr 5125 Btwn cbtwn 27935 Colinear ccolin 34732 OutsideOfcoutsideof 34814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3461 df-dif 3931 df-br 5126 df-outsideof 34815 |
This theorem is referenced by: broutsideof2 34817 outsideofrflx 34822 outsidele 34827 outsideofcol 34828 |
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