Enderton 2.2.17

Consider a language with equality whose only parameter (aside from $\displaystyle \forall$) is a two-place predicate symbol $\displaystyle P$. Show that if $\displaystyle A$ is finite and $\displaystyle A \equiv B$ ($\displaystyle A$ is elementary equivalent to $\displaystyle B$), $\displaystyle A$ is isomorphic to $\displaystyle B$.

Suggestion: Suppose the universe of $\displaystyle A$ has size $\displaystyle n$. Make a single sentence $\displaystyle \sigma$ of the form $\displaystyle \exists v_1 \cdots \exists v_n \theta$ that describes $\displaystyle A$ "completely". That is, on the one hand, $\displaystyle \sigma$ must be true in $\displaystyle A$. And on the other hand, any model of $\displaystyle \sigma$ must be exactly like (i.e., isomorphic to) $\displaystyle A$.

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To get started, how do I make a single sentence $\displaystyle \sigma$ of the form $\displaystyle \exists v_1 \cdots \exists v_n \theta$ that describes $\displaystyle A$ "completely"?

Thanks.