$(4, 5)$
$(4, 6)$
$(0, 4)$
$(5, \infty)$
$(5, 6)$

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Pretty straightforward. Since none are true, we can take the reverse of each statement.

We are less than $6$ miles away, so $d < 6$ .

We are more than $5$ miles away, so $d > 5$ .

We are more than $4$ miles away, so $d > 4$ .

The union of these statements is $5 < d < 6 \implies \boxed{(5, 6)}$ .