Set of 3 Integers each Divisor of Sum of Other Two

Theorem
There exists exactly one set of positive integers such that each is a divisor of the sum of the other two:
 * $\set {1, 2, 3}$

Proof
We have that:
 * $5 \times 1 = 2 + 3$ so $1 \divides 2 + 3$
 * $2 \times 2 = 1 + 3$ so $2 \divides 1 + 3$
 * $1 \times 3 = 1 + 2$ so $3 \divides 1 + 2$

It remains to be shown that this is the only such set.