My proof that the empty set is unique

Question

I'm trying to prove that the empty set is unique.

Proof:

Let $U = \{ a \}$ be the universal set.

Assume $a \not\in \emptyset '$ and $a \not\in \emptyset$.

Without loss of generality, since $a \not\in \emptyset'$, $\emptyset '$ does not contain any elements. Since $\emptyset '$ does not contain any elements, it must by default be a subset of $\emptyset$, since the conditional statement

$$a \in \emptyset ' \Rightarrow \emptyset ' \subseteq \emptyset$$

is vacuously true.

Therefore, since $\emptyset' \subseteq \emptyset$ and $\emptyset \subseteq \emptyset '$, we have that $\emptyset ' = \emptyset$. $\tag*{$\blacksquare$}$

I would appreciate it if people could please provide feedback as to the correctness of my proof.

EDIT: Please be specific about what is incorrect and why. That way, I can learn what I did wrong and improve much more effectively.

Answer 1

The empty set is a subset of any set. Let $A$ and $B$ be two empty sets. Since $A$ is empty, then $A \subseteq B$. Similarly, $B \subseteq A$. Hence $A=B$.

EDIT: Your assumptions are a bit suspicious and the use of the universal set is really unnecessary. Basically the part: assume $a\notin \emptyset'$ and $\emptyset'$ does not contain any elements is a bit wordy and I am not sure if it is a valid logic flow. The rest of your solution is pretty much the idea that I uncover above. All you need to claim is that two sets are empty and then use the fact that they are subsets of each-other.

Answer 2

Let $A$ and $B$ be two empty sets. Then the assertions $x\in A$ and $x\in B$ are logically equivalent. By the definition of equality of sets, $A=B$ iff $\forall x(x\in A\Longleftrightarrow x\in B)$, it follows that $A=B$.