Step |
Hyp |
Ref |
Expression |
1 |
|
btwnconn1 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) -> ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) ) ) |
2 |
|
simpr2 |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) -> B Btwn <. A , C >. ) |
3 |
2
|
anim1i |
|- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ C Btwn <. A , D >. ) -> ( B Btwn <. A , C >. /\ C Btwn <. A , D >. ) ) |
4 |
|
btwnexch3 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ C Btwn <. A , D >. ) -> C Btwn <. B , D >. ) ) |
5 |
4
|
ad2antrr |
|- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ C Btwn <. A , D >. ) -> ( ( B Btwn <. A , C >. /\ C Btwn <. A , D >. ) -> C Btwn <. B , D >. ) ) |
6 |
3 5
|
mpd |
|- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ C Btwn <. A , D >. ) -> C Btwn <. B , D >. ) |
7 |
6
|
ex |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) -> ( C Btwn <. A , D >. -> C Btwn <. B , D >. ) ) |
8 |
|
simpr3 |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) -> B Btwn <. A , D >. ) |
9 |
|
simp3r |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) ) |
10 |
|
simp3l |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) ) |
11 |
9 10
|
jca |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( D e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) |
12 |
|
btwnexch3 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , C >. ) -> D Btwn <. B , C >. ) ) |
13 |
11 12
|
syld3an3 |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , C >. ) -> D Btwn <. B , C >. ) ) |
14 |
13
|
adantr |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) -> ( ( B Btwn <. A , D >. /\ D Btwn <. A , C >. ) -> D Btwn <. B , C >. ) ) |
15 |
8 14
|
mpand |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) -> ( D Btwn <. A , C >. -> D Btwn <. B , C >. ) ) |
16 |
7 15
|
orim12d |
|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) -> ( ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) -> ( C Btwn <. B , D >. \/ D Btwn <. B , C >. ) ) ) |
17 |
16
|
ex |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) -> ( ( C Btwn <. A , D >. \/ D Btwn <. A , C >. ) -> ( C Btwn <. B , D >. \/ D Btwn <. B , C >. ) ) ) ) |
18 |
1 17
|
mpdd |
|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( A =/= B /\ B Btwn <. A , C >. /\ B Btwn <. A , D >. ) -> ( C Btwn <. B , D >. \/ D Btwn <. B , C >. ) ) ) |