Step |
Hyp |
Ref |
Expression |
1 |
|
ltnelicc.a |
|- ( ph -> A e. RR ) |
2 |
|
ltnelicc.b |
|- ( ph -> B e. RR* ) |
3 |
|
ltnelicc.c |
|- ( ph -> C e. RR* ) |
4 |
|
ltnelicc.clta |
|- ( ph -> C < A ) |
5 |
1
|
rexrd |
|- ( ph -> A e. RR* ) |
6 |
|
xrltnle |
|- ( ( C e. RR* /\ A e. RR* ) -> ( C < A <-> -. A <_ C ) ) |
7 |
3 5 6
|
syl2anc |
|- ( ph -> ( C < A <-> -. A <_ C ) ) |
8 |
4 7
|
mpbid |
|- ( ph -> -. A <_ C ) |
9 |
8
|
intnanrd |
|- ( ph -> -. ( A <_ C /\ C <_ B ) ) |
10 |
|
elicc4 |
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) ) |
11 |
5 2 3 10
|
syl3anc |
|- ( ph -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) ) |
12 |
9 11
|
mtbird |
|- ( ph -> -. C e. ( A [,] B ) ) |