Show that if $\mathcal{M}$ is a model of $\Gamma$, and $\Delta \subseteq \Gamma$, then $\mathcal{M}$ is also a model of $\Delta$.

Question

Show that if $\mathcal{M}$ is a model of $\Gamma$, and $\Delta \subseteq \Gamma$, then $\mathcal{M}$ is also a model of $\Delta$.

Answer 1

1. First, note that $\Gamma$ and $\Delta$ are sets of sentences.
2. Let's assume that $\mathcal{M}$ is a model of $\Gamma$, this means that $\mathcal{M} \models \varphi$ for every sentence $\varphi \in \Gamma$
3. Since $\Delta \subseteq \Gamma$, and $\Delta$ is a set of sentences, by def. $\subseteq$, every member of $\Delta$ is also a member of $\Gamma$
4. Then $\mathcal{M} \models \varphi$ for every $\varphi \in \Delta$
5. Therefore, $\mathcal{M}$ is a model of $\Delta$