Description: Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eliood.1 | |- ( ph -> A e. RR* ) |
|
eliood.2 | |- ( ph -> B e. RR* ) |
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eliood.3 | |- ( ph -> C e. RR ) |
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eliood.4 | |- ( ph -> A < C ) |
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eliood.5 | |- ( ph -> C < B ) |
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Assertion | eliood | |- ( ph -> C e. ( A (,) B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliood.1 | |- ( ph -> A e. RR* ) |
|
2 | eliood.2 | |- ( ph -> B e. RR* ) |
|
3 | eliood.3 | |- ( ph -> C e. RR ) |
|
4 | eliood.4 | |- ( ph -> A < C ) |
|
5 | eliood.5 | |- ( ph -> C < B ) |
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6 | elioo2 | |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A (,) B ) <-> ( C e. RR /\ A < C /\ C < B ) ) ) |
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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 ) ) |