Description: Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eliccxrd.1 | |- ( ph -> A e. RR* ) |
|
eliccxrd.2 | |- ( ph -> B e. RR* ) |
||
eliccxrd.3 | |- ( ph -> C e. RR* ) |
||
eliccxrd.4 | |- ( ph -> A <_ C ) |
||
eliccxrd.5 | |- ( ph -> C <_ B ) |
||
Assertion | eliccxrd | |- ( ph -> C e. ( A [,] B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccxrd.1 | |- ( ph -> A e. RR* ) |
|
2 | eliccxrd.2 | |- ( ph -> B e. RR* ) |
|
3 | eliccxrd.3 | |- ( ph -> C e. RR* ) |
|
4 | eliccxrd.4 | |- ( ph -> A <_ C ) |
|
5 | eliccxrd.5 | |- ( ph -> C <_ B ) |
|
6 | 4 5 | jca | |- ( ph -> ( A <_ C /\ C <_ B ) ) |
7 | elicc4 | |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) ) |
|
8 | 1 2 3 7 | syl3anc | |- ( ph -> ( C e. ( A [,] B ) <-> ( A <_ C /\ C <_ B ) ) ) |
9 | 6 8 | mpbird | |- ( ph -> C e. ( A [,] B ) ) |