Handling vacuous truths in proving elementary set theory So I was working on this simple problem:

For sets $A$ and $B$, prove that $A\subset B\Leftrightarrow A\cup B=B$.

Now I know how this proof works: First suppose $A\subset B$, then take $x\in A\cup B$, and so on. What I'm concerned about is how can we handle vacuous truths(or more specifically, empty sets). 
For example, let's show that if $A\cup B=B$, then $A\subset B$. We will most likely take $x\in A$. However, if $A=\varnothing$, then there is no such $x$. Another example is when we show the converse. Then, we will take $x\in A\cup B$. However, there is no such $x$ if $A\cup B$ is empty.
I know this may sound like nitpicking, but I'm not sure if I can just ignore those cases. I've seen numerous proofs here on MSE, but they seem to have no problem taking an element from a set that may be empty. Can anyone explain this to me? Thanks in advance.

You are correct, and formally a lot of proofs should start with: If $A = \emptyset$, then it holds that $P$. So we assume that $A \neq \emptyset$ [..]
However in most situations the claim follows immediately for empty sets (because of vacuous truths), so it is common practice to omit the case $A = \emptyset$.
