For A,B,C,and D sets. Is it true that $(A - C) \times (B - D) \subset (A \times B) - (C \times D)$ proof verification

Question

I claim that $(A - C) \times (B - D) \subset (A \times B) - (C \times D)$.

Let $(x,y) \in (A - C) \times (B - D)$. It means that $x \in (A - C)$ and $y \in (B - D)$. This implies that $x \in A$ and $y \in B$ such that $x$ and $y$ doesn't exist in C and D respectively. Therefore $(A - C) \times (B - D) \subset (A \times B) - (C \times D)$.

I haven't found counter example yet but it should be true that this is strict inclusion.

Answer 1

With $C = \emptyset$, and $A,B,D$ non-empty with $B = D$, you get

$(A \setminus C) \times (B \setminus D) = \emptyset \subsetneq A \times B = (A \times B) \setminus (C \times D)$