Critique my proof of: Suppose $F$ and $G$ are families of sets, and $F \cap G \neq \varnothing$. Then $\bigcap F \subseteq \bigcup G$.

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.

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 $A_0$".

I would say two things are missing from the proof the way you wrote it:

1. You are proving that $\bigcap F \subseteq \bigcup G$, and you start with "Let $x \in \bigcap F$." That's a good start. I'm now expecting you to prove that $x \in \bigcup G$, but you never asserted that conclusion. You were almost there; you had $A_0 \in G$ and $x \in A_0$, which is all you need to conclude that $x \in \bigcup G$, but you never actually asserted that conclusion. So you need to put "Therefore $x \in \bigcup G$" after "$A_0 \in G$ and $x \in A_0$."

2. You didn't justify your claim that $x \in A_0$. The claim is correct, but you should give a reason. The reason is that $x \in \bigcap F$ and $A_0 \in F$.