 Solution

This class is axiomatisable in the original language. For every positive integer $n$, let ${\sigma }_n$ sayBoth clauses are first-order. If one of the two sets is infinite, every ${\sigma }_n$ holds. Conversely, if both sets were finite, choosing $n$ larger than both cardinalities would make ${\sigma }_n$ false. Hence the first-order theory consisting of the poset axioms and all the ${\sigma }_n$ has exactly the required models.