Using the definition of a delta elementary class ($EC_\Delta$ ) to be the set of structures which entail every formula in some set $\Gamma$ . We define an elementary class (EC) to be such a class generated by a finite $\Gamma$.
Suppose $\mathcal K$ is a class of $EC_\Delta$ classes—that is, the elements of $\mathcal{K}$ are $EC_\Delta$ classes of models— and suppose $\mathcal{B}$ is EC and that $\mathcal \bigcap \mathcal{K} \subseteq \mathcal{B}$. How to show that there is a finite subcollection $\mathcal{K}′$ of $\mathcal{K} $ (i.e., a finite collection of $EC_\Delta$ classes in $\mathcal K $) such that $\mathcal \bigcap \mathcal{K}′ \subseteq \mathcal{B}$.
Each class $K\in \mathcal{K}$ is $EC_\Delta$, so it is axiomatized by a theory $T_K$. And $\mathcal{B}$ is $EC$, so it is axiomatized by a sentence $\varphi$.
Then $\bigcup_{K\in \mathcal{K}} T_K$ axiomatizes $\bigcap \mathcal{K}$, and to say that $\bigcap \mathcal{K}\subseteq \mathcal{B}$ is to say that $\bigcup_{K\in \mathcal{K}} T_K\models \varphi$. By compactness, there are finitely many sentences $\psi_1,\dots,\psi_n\in \bigcup_{K\in \mathcal{K}} T_K$ such that $\{\psi_1,\dots,\psi_n\}\models \varphi$.
For all $1\leq i\leq n$, $\psi_i$ is in $T_{K_i}$ for some $K_i\in \mathcal{K}$. Let $\mathcal{K}' = \{K_1,\dots,K_n\}$. Then $\bigcap\mathcal{K}'\subseteq \mathcal{B}$.
Why? $M\in \bigcap\mathcal{K}'$ $\implies$ $\forall i, M\in K_i$ $\implies$ $\forall i, M\models T_{K_i}$ $\implies$ $\forall i, M\models \psi_i$ $\implies$ $M\models \varphi$ $\implies$ $M\in \mathcal{B}$.