I haven't been able to figure out if the following reasoning is correct, so I'd like to have the opinion of other people on that.
The goal is to prove that the empty set is unique.
In order to do that, let $E$ be a set, $A$ be an empty set and $B$ be an empty set. I want to prove that $A = B$.
First, I can try proving that $A \subset B$.
I know that $\forall x \in E, x \notin A$. Now I can consider this proposition : $x \in A \Rightarrow x \in B$
Since $x \notin A$, I can safely say that that the proposition above is true.
Hence, I can conclude that $A \subset B$
Am I right in my conclusion?