Let $S$ be any set of axioms extending ZF. Suppose $\phi_1, ... \phi_n \in S$. Show that, if from $\phi_1, ... \phi_n$ one can prove all axioms of $S$, then $S$ is inconsistent. Conclude that ZF is not finitely axiomatizable.  So in order to show that $S$ is inconsistent, I need to show $S \vdash (\phi_1 \land ... \land\phi_n )$ and $S \vdash \lnot (\phi_1 \land ... \land\phi_n )$. But how do I do this and how do I conclude that ZF is not finitely axiomatizable?
As we know a first order theory $T$ is finitely axiomatizable if there is a finite set $F\subseteq T$ of axioms such that $F\vdash \sigma$ for every $\sigma \in T$.   How we can prove if $\mathsf{ZF}$ is consistent, then $\mathsf{ZF}$ is not finitely axiomatizable? By using the Reflection Theorem or any other if we could use?
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