A simple proof that elementary equivalence and isomorphism coincide for finite structures?

Question

I'm wondering if there's a straightforward proof of this result I've seen mentioned in quite a few places. If $\mathcal{L}$ is finite, of course, this is trivial since there's a single formula that defines the structure up to isomorphism. But what if $\mathcal{L}$ is infinite?

Answer 1

If the two finite structures $\mathcal A$ and $\mathcal B$ are elementarily equivalent, then they have the same cardinality. Now, there are just a finite number of bijections from $A$ to $B.$ Assume for a contradiction that none of those bijections is an isomorphism. For each bijection, choose an $\mathcal L$-relation which is not preserved. Consider the finite sublanguage $\mathcal L_0$ of $\mathcal L$ consisting of the chosen relations. Now, the $\mathcal L_0$-reducts of $\mathcal A$ and $\mathcal B$ are elementarily equivalent, and therefore isomorphic; but this is impossible because of the way $\mathcal L_0$ was chosen.