Prove using set identities

Question

Let $A$,$B$,$C$ be sets. Draw a Venn diagram and color the region $(A−C) \cap (C−B)$. Prove that $(A−C)\cap (C−B)=∅$.

Already done the first part, but the proving part I am stuck at. I have no clue on how to do this. Please show/guide me

Answer 1

We have $$\begin{align} x\in (A-C)\cap(C-B)&\iff (x\in A-C)\text{ and }(x\in C-B) \\ &\iff (x\in A \text{ and }x\notin C) \text{ and }(x\in C\text{ and }x\notin B)\\ &\iff \color{red}{(x\in C\text{ and }x\notin C)}\text{ and }(x\in A\text{ and }x\notin B), \end{align}$$

which is a $\color{red}{\text{contradiction}}$.

Hence $(A-C)\cap(C-B)=\varnothing $.