If A is a subset of B and C is a subset of D, then A intersect C is a subset of B intersect D.
Please any help in proving this conditional would be much appreciated
Suppose that $\displaystyle x\in{A\cap\\C}$. Then, $\displaystyle x\in{A}$ and $\displaystyle x\in{C}$. Since $\displaystyle A\subset{B}$, $\displaystyle x\in{B}$ and, because $\displaystyle C\subset{D}$, $\displaystyle x\in{D}$ as well. Thus, $\displaystyle x\in{B\cap{D}}$.