Prove that any relation that is both an equivalence relation and a partial ordering is the identity relation. That is, if X is a set and R is a relation on X that is both a partial ordering and an equivalence relation, then aRb if and only if a=b.
Is this logic sound?
By definition of transitivity, if (a,b) exists and (b,c) exists, then (a,c) exists. However we know this is also symmetric which means (c,a) exists. By definition of antisymmetry, if those two ordered pairs exist, then a = c.
Also by reflexivity, (a,a) exists for all a, and in these ordered pairs, b = a.