Hi so I want to know if my proof attempt is correct, so here's the question:
Suppose that $F$ is a family of sets. Prove that if $\varnothing\in F$ then $\cap F=\varnothing$.
Suppose $\varnothing\in F$ and suppose $\cap F\ne\varnothing$. But this means that $x\in\varnothing$, since $\varnothing\in F$. Since $A$ was an arbitrary element of $F$, $x\in\varnothing$. But this is a contradiction since $\varnothing$ is the empty set. Thus $\cap F=\varnothing$.
Therefore if $\varnothing\in F$ then $\cap F=\varnothing$.