Prove:

For all sets A, B, and C, if A ⊆ B and B ⊆ Cc, then A ∩ C = ∅.

Step-by-step explanation:

Let's take "a" an element from A, a ⊆ A.

As A ⊆ B, a ⊆ A ⊆ B, so a ⊆ B.

Therefore, a ⊆ B ⊆ Cc, a ⊆ Cc.

Let's remember that Cc is exactly the opposite of C. That means that an element is in C or in Cc; it has to be in one of them but not in both.

As a ⊆ Cc, a ⊄ C.

As we can generalize this for every element of A, there is not element of A that is contained in C.

Therefore, the intersection (the elements that are in both A and C) is empty.