Empty Set is Unique/Proof 2

Theorem

The empty set is unique.

Proof

Let $A$ and $B$ both be empty sets.

Thus:

$\forall x: \neg \paren {x \in A}$

and:

$\forall x: \neg \paren {x \in B}$

Hence:

$x \notin A \iff x \notin B$

and so:

$x \in A \iff x \in B$

vacuously.

From the Axiom of Extension:

$\forall x: \paren {x \in A \iff x \in B} \iff A = B$

Hence the result.

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 3$: Unordered Pairs
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): empty: 1.
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets: Exercise $1.2.4$
• 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $4$