TITLE: How to prove set equivalence with an empty set?
QUESTION [5 upvotes]: If I want to prove $A = \emptyset$ for some set A, how would I go?
My first thoughts were to prove it like any set equivalence, by assuming that $x \in \emptyset$ which means that $x \in \emptyset$ or $x \in A$ implying $x \in \emptyset \cup A = A$. But I would not know how to go from $y \in A$ to $y \in \emptyset$ which is impossible obviously.
I also thought about trying $x \in A$ and try to prove $x \notin A$. Will this imply that $A = \emptyset$?

REPLY [2 votes]: From the definition of equality of sets you need to show that,
$$A\subseteq \emptyset \land \emptyset\subseteq A$$
If any of them is false then the statement cannot be true. Prove that the first assertion is false.

To prove this just assume that $\exists x\in A$. Then this would imply that $x\in \emptyset$ but that is false.