Step |
Hyp |
Ref |
Expression |
1 |
|
btwncom |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Btwn <. B , C >. <-> A Btwn <. C , B >. ) ) |
2 |
|
3anrot |
|- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) |
3 |
|
btwncom |
|- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) ) |
4 |
2 3
|
sylan2b |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( B Btwn <. C , A >. <-> B Btwn <. A , C >. ) ) |
5 |
|
3anrot |
|- ( ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) |
6 |
|
btwncom |
|- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. <-> C Btwn <. B , A >. ) ) |
7 |
5 6
|
sylan2br |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. <-> C Btwn <. B , A >. ) ) |
8 |
1 4 7
|
3orbi123d |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) <-> ( A Btwn <. C , B >. \/ B Btwn <. A , C >. \/ C Btwn <. B , A >. ) ) ) |
9 |
|
3orcomb |
|- ( ( A Btwn <. C , B >. \/ B Btwn <. A , C >. \/ C Btwn <. B , A >. ) <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) |
10 |
8 9
|
syl6bb |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) ) |
11 |
|
brcolinear |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) ) ) |
12 |
|
3ancomb |
|- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) |
13 |
|
brcolinear |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Colinear <. C , B >. <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) ) |
14 |
12 13
|
sylan2b |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. C , B >. <-> ( A Btwn <. C , B >. \/ C Btwn <. B , A >. \/ B Btwn <. A , C >. ) ) ) |
15 |
10 11 14
|
3bitr4d |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A Colinear <. B , C >. <-> A Colinear <. C , B >. ) ) |