In first-order logic, an immediate corollary of the definitions of the theory of a set of models M (denoted $Th(M)$) and the Model of a set of sentences S (denoted $Mod(S)$) is:
$M\subseteq Mod(Th(M)) $
But how is this proven? (for example, what happens if $Th(M)=\emptyset$ and $M\neq \emptyset$)
$Th(M)$ is the set of all sentences true in $M$. Take $A \in M$ and $\phi$ such that $A \vDash \phi$, then $\phi \in Th(M)$. By definition, $A \in Mod(Th(M))$.