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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 | ⊢ (𝑃OutsideOf〈𝐴, 𝐵〉 ↔ (𝑃 Colinear 〈𝐴, 𝐵〉 ∧ ¬ 𝑃 Btwn 〈𝐴, 𝐵〉)) |
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〉)) |
Colors of variables: wff setvar class |
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 |
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