= Solution
This class is not <first-order axiomatizable class>[axiomatisable]. Suppose that a theory $T$ axiomatized it. Add constants $m_0,m_1,\ldots$, require each $m_i$ to be a <maximal element of a partially ordered set>[maximal element], and require $m_i\ne m_j$ for $i\ne j$. Every finite part of this enlarged theory has a model: choose a finite <partially ordered set> with enough maximal elements. By the <compactness theorem> the whole theory has a model, but its reduct is a model of $T$ with infinitely many maximal elements, a contradiction. This is the <compactness obstruction to axiomatizing finitely many maximal elements>.
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