Show that the empty set is uniqueSolutionLet x and y be empt

Solution

Let x and y be empty sets, then u y and u x are always false for all sets u. Thus (u yu x) is true for all sets u and since by the axiom of equality (u(u xu y)) (x = y) is true then it follows that (x = y) must be true.

or

Let and both be empty sets.

From Empty Set is Subset of All Sets, , because is empty.

Likewise, we have , since is empty.

Together, by the definition of set equality, this implies that =.

Thus there is only one empty set.