How can you prove this mathematical conjecture?

Question

Let A, B, and C be subsets of the universal set U. if A∩compliment of B=A,B – C=B, and (AUB) ∩C=C, then C is in the subset of A.

Answer 1

Let x∈C. Since C = (A∪B)∩C, it follows that x∈(A∪B)∩C and thus x∈A∪B. However, since x∈C, x∉B\C, so if x were an element of B, it would follow that B≠B\C, a contradiction. Thus x∈A∪B and x∉B, so x∈A. Since this is true for an arbitrary x∈C, it follows that C⊆A.