Inclusion in sequences of theories and models

Question

Is this theorem true, and if so does it have a name and where can I reference it?

"Let $T_1$ and $T_2$ be theories where $T_1 \subset T_2$. If $K_1$ and $K_2$ are the classes of all models of $T_1$ and $T_2$ respectively, then $K_1 \supseteq K_2$."

Answer 1

The fact that $T_1 \subseteq T_2 \Rightarrow K_1 \supseteq K_2$ follows almost immediately from the definition of what it is to be a model of a theory: to say that $T_1 \subseteq T_2$ means that $T_2$ requires at least as many things to be true than $T_1$ does. This means that a model of $T_2$ is automatically a model of $T_1$, so that $K_2 \subseteq K_1$.