Let L be an arbitrary language and $T_1$ and $T_2$ be non-empty L-theories. Suppose that the L-theory $T_1 \cup T_2$ is inconsistent. Proof that there is an L-sentence $\phi$ such that $T_1$ $\vDash$ $\phi$ and $T_2$ $\vDash$ $\neg$ $\phi$.
Is the following proof correct?
Because of the compactness theorem, there exist two subtheories $\Delta_1$ $\subset$ $T_1$ and $\Delta_2$ $\subset$ $T_2$ such that $\Delta_1$ $\cup$ $\Delta_2$ $\models$ $\bot$ $\equiv$ $\neg$ $\phi$ $\wedge$ $\phi$. So we found our sentence?