Just wonder if my working is right. Let $(A_{\alpha})_{\alpha \in I}$ be an indexed family of subsets of a set $S$. Let $J \subset I$. Want to show : $\cap_{\alpha \in J} A_{\alpha} \supset \cap_{\alpha \in I} A_{\alpha}$
Attempts: First of all, if we reduce it to the case of 3 sets we see that the above relation holds (draw Venn diagrams...) Let $x \in \cap_{\alpha \in I} A_{\alpha}$. Then we have $x \in A_{\alpha} \forall \alpha \in I$. Since $J \subset I$, we have $x \in A_{\alpha} \forall \alpha \in J$.
I think my argument is not solid though, even if the relation to be shown just seems kind of trivial.