relation between union and intersection A and B are any two sets.
if A U B ≠ B
is it also true that A ∩ B ≠ A ?
and if so then why?
This is just a step I am using for a bigger proof, if the above is not true then I'll have to search for a different direction.
Thanks in advance.

Yes, it is true. $A \cup B \neq B$ means $\exists x \in A \setminus B$ (otherwise, we would have $ A \subset B$ which implies $A \cup B = B$).
And since $x \not\in B$, $x \not \in A \cap B$ while $x \in A$
