![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 | ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-outsideof 34920 | . . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn ) | |
2 | 1 | breqi 5147 | . 2 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ 𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉) |
3 | brdif 5194 | . 2 ⊢ (𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∖ cdif 3941 〈cop 4628 class class class wbr 5141 Btwn cbtwn 28012 Colinear ccolin 34837 OutsideOfcoutsideof 34919 |
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 3475 df-dif 3947 df-br 5142 df-outsideof 34920 |
This theorem is referenced by: broutsideof2 34922 outsideofrflx 34927 outsidele 34932 outsideofcol 34933 |
Copyright terms: Public domain | W3C validator |