Step |
Hyp |
Ref |
Expression |
1 |
|
eliccelicod.a |
|- ( ph -> A e. RR* ) |
2 |
|
eliccelicod.b |
|- ( ph -> B e. RR* ) |
3 |
|
eliccelicod.c |
|- ( ph -> C e. ( A [,] B ) ) |
4 |
|
eliccelicod.d |
|- ( ph -> C =/= B ) |
5 |
|
eliccxr |
|- ( C e. ( A [,] B ) -> C e. RR* ) |
6 |
3 5
|
syl |
|- ( ph -> C e. RR* ) |
7 |
|
iccgelb |
|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> A <_ C ) |
8 |
1 2 3 7
|
syl3anc |
|- ( ph -> A <_ C ) |
9 |
|
iccleub |
|- ( ( A e. RR* /\ B e. RR* /\ C e. ( A [,] B ) ) -> C <_ B ) |
10 |
1 2 3 9
|
syl3anc |
|- ( ph -> C <_ B ) |
11 |
6 2 10 4
|
xrleneltd |
|- ( ph -> C < B ) |
12 |
1 2 6 8 11
|
elicod |
|- ( ph -> C e. ( A [,) B ) ) |