The empty set

#empty set#uniqueness#axioms

Statement

Show that the empty set is unique.

Hint

Suppose $\emptyset$ and $\emptyset'$ are both empty; show they are equal.

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Argument. Two sets are equal iff they have the same elements. Since both are empty, they have no elements. So they are equal.

Conclusion. There is only one empty set, denoted $\emptyset$ .