 Solution

The Godel completeness theorem says that a first-order sentence follows semantically from a theory exactly when it is formally derivable from it. The compactness theorem says that a theory has a model exactly when every finite subset has a model.

Fix $i$. For each $j\ne i$, the theory $T_i\cup T_j$ has no model because the model classes form a partition. By compactness, some finite ${\Delta }_{ij}\subseteq T_i$ is already inconsistent with $T_j$. Leta finite subset of $T_i$. Every model of $T_i$ models ${\Delta }_i$. Conversely, a model of ${\Delta }_i$ belongs to exactly one class $T_j$; if $j\ne i$, it would model both $T_j$ and ${\Delta }_{ij}$, a contradiction. Hence it belongs to $T_i$, and ${\Delta }_i$ finitely axiomatizes $T_i$.