Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | btwncolinear6 | |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> C Colinear <. B , A >. ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwncolinear1 | |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> A Colinear <. B , C >. ) ) |
|
2 | colinearperm5 | |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> C Colinear <. B , A >. ) ) |
|
3 | 1 2 | sylibd | |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> C Colinear <. B , A >. ) ) |