I think this is correct, and this explanation works regardless whether A has other elements besides 1, 2. Any other elements, whether their set is empty or not, must be in B. (Recall that the empty set is a subset of any set.)First case, if A has other elements that are not 1, 2, then those elements must be in B since A-B is a subset of {1, 2}. Since B - C is a subset of {1, 2}, those elements must also be in C. Thus, A-C would remove the elements that aren't 1 or 2.

A similar, but more formal, way to write this is the following.

Lemma. .

Proof. ...

Proof of transitivity. Suppose and . By Lemma, this means and . Therefore, , which, again by Lemma, means that .