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 33576 | . . 3 ⊢ OutsideOf = ( Colinear ∖ Btwn ) | |
2 | 1 | breqi 5064 | . 2 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ 𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉) |
3 | brdif 5111 | . 2 ⊢ (𝑃( Colinear ∖ Btwn )〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) | |
4 | 2, 3 | bitri 277 | 1 ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∖ cdif 3932 〈cop 4566 class class class wbr 5058 Btwn cbtwn 26669 Colinear ccolin 33493 OutsideOfcoutsideof 33575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-br 5059 df-outsideof 33576 |
This theorem is referenced by: broutsideof2 33578 outsideofrflx 33583 outsidele 33588 outsideofcol 33589 |
Copyright terms: Public domain | W3C validator |