Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | icossioo | |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A (,) B ) ) |
Step | Hyp | Ref | Expression |
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1 | df-ioo | |- (,) = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a < x /\ x < b ) } ) |
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2 | df-ico | |- [,) = ( a e. RR* , b e. RR* |-> { x e. RR* | ( a <_ x /\ x < b ) } ) |
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3 | xrltletr | |- ( ( A e. RR* /\ C e. RR* /\ w e. RR* ) -> ( ( A < C /\ C <_ w ) -> A < w ) ) |
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4 | xrltletr | |- ( ( w e. RR* /\ D e. RR* /\ B e. RR* ) -> ( ( w < D /\ D <_ B ) -> w < B ) ) |
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5 | 1 2 3 4 | ixxss12 | |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < C /\ D <_ B ) ) -> ( C [,) D ) C_ ( A (,) B ) ) |