Description: Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eliocd.a | |- ( ph -> A e. RR* ) |
|
eliocd.b | |- ( ph -> B e. RR* ) |
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eliocd.c | |- ( ph -> C e. RR* ) |
||
eliocd.altc | |- ( ph -> A < C ) |
||
eliocd.cleb | |- ( ph -> C <_ B ) |
||
Assertion | eliocd | |- ( ph -> C e. ( A (,] B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliocd.a | |- ( ph -> A e. RR* ) |
|
2 | eliocd.b | |- ( ph -> B e. RR* ) |
|
3 | eliocd.c | |- ( ph -> C e. RR* ) |
|
4 | eliocd.altc | |- ( ph -> A < C ) |
|
5 | eliocd.cleb | |- ( ph -> C <_ B ) |
|
6 | elioc1 | |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) ) |
|
7 | 1 2 6 | syl2anc | |- ( ph -> ( C e. ( A (,] B ) <-> ( C e. RR* /\ A < C /\ C <_ B ) ) ) |
8 | 3 4 5 7 | mpbir3and | |- ( ph -> C e. ( A (,] B ) ) |