The proof wants the statement to be proved for general sets A, B, and C; it is not asking for A, B, and C to be specific sets. The goal of the proof is to show that A⊆C. By definition A⊆C if for each element x in A, x is an element in C. To proceed with the proof, pick any arbitrary element x' in A. Our goal is to show that x' is in C. If we can show that any arbitrary element in A is also in C, then it would follow that each element in A is also in C, which satisfies the definition of A⊆C.

By hypothesis, We are given that A⊆B, so by definition, for each element y in A, y is in B. We are also given that B⊆C, so by definition, for each element z in B, z is an element in C. Consequently, since x' is in A, then it follows that x' is in B, and since x' is in B, x' is also in C. Since x' is an arbitrary element in A and x' is in C, then it follows that each element in A is also in C. Therefore, we have satisfied the definition of A⊆C, so A⊆C.