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(i) There is at least one natural number.

(ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that $y$ stands in the

successor relation to $x$.

(iii) No two natural numbers have the same successor.

(iv) There is a natural number, namely 0, that is not the successor of any number.

Bearing these facts in mind, what's a formula in first-order logic that that is satisfiable by a valuation only if the domain of the valuation is infinite. Contain some non-logical vocabulary in presentation of course.