I guess this should be a proof?

Logically:

If everything that's in A is in B U C, that means that everything that is in A is either in B or C (or both).

Now, if you take everything that's in A and throw out everything that is in B, the only thing that's left is everything that was in C. It was all in B and C to begin with, and now you're eliminating B.

Here's an attempt at a slightly more formal proof - I haven't done this kind of stuff in a while, but again it seems to make sense. To me, anyway.

Assume A is a subset of B U C. Let x be in A - B. Then x is in A by definition. Also, by the definition of A - B, x is NOT in B. Also, by our assumption (the hypothesis), x is in B U C (also by definition of subset, as A is a subset of B U C). Thus, x is in B or x is in C or both. But we already assumed x was in A - B, so x was not in B. Then x must be in C. Therefore, if A is a subset of B U C and x is in A - B, then x is in C. Thus, A - B is a subset of C.