Ultraproduct proof of the Ehrenfeucht-Mostowski theorem

Expand an infinite first-order structure by Skolem functions, choose distinct elements $b_n$, and fix a nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$. For each finite subset $F$ of the required index order, form the ultrapower over ${\mathbb{N}}^F$ using the ordered Fubini product of ultrafilters. Pullback along coordinate projections gives coherent elementary maps by the Łoś theorem. In their directed limit of elementary embeddings, the classes of the coordinate functions $\mathbf{n}\mapsto b_{n_i}$ are distinct and form an order-indiscernible sequence; every first-order formula on increasing coordinates has the same nested ultrafilter test. Their Skolem hull gives an Ehrenfeucht-Mostowski model. An index-order automorphism sends a term in the generators to the same term in the transported generators; indiscernibility makes this well defined. Uniqueness is in the specified Skolem expansion, not necessarily in its reduct. If generators must respect a definable infinite order, first produce an increasing sequence by an ultrapower of finite increasing chains.