Critique my proof on correctness, structure, etc. Any help is much appreciated!
Theorem. Suppose $F$ and $G$ are families of sets, and $F \cap G \neq \varnothing$. Then $\bigcap F \subseteq \bigcup G$.
Proof. Let $x \in \bigcap F$. Because $F \cap G \neq \varnothing$, we can let $A_{0} \in F \cap G$. Thus, $A_{0} \in F$, $A_{0} \in G$, and $x \in A_{0}$. Because x is arbitrary, we can conculde that $\forall x(x \in \bigcap F \implies x \in \bigcup G$), so $\bigcap F \subseteq \bigcup G$.
I feel like my proof is missing some things, but I'm not sure what.
Best Answer
This is a fine proof and I would give it full credit if I were grading a course.
To more cleanly demonstrate the chain of reasoning, consider breaking the sentence starting "Thus" into two sentences:
Since $A_0 \in F \cap G,$ we know $A_0 \in F$ and therefore $x \in A_0$. Further, since $A_0 \in G$, $A_0 \subset \bigcup G$, so $x \in \bigcup G$.
You might also change the phrase "we can let $A_0$.." to "there exists some $A_0$".