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.

It's easier to prove this the other way round, i.e. you can show that 

if $A\cap B=A$, then $A\cup B=B$.

This is easy to show. The fact that $A\cap B = A$ implies that $A\subseteq B$, which also directly means that $A\cup B = B$.
Or, if you want the traditional long way round:


*

*Let $b\in A\cup B$.


*

*Then, if $b\in B$, we are done

*If $b\in A$, then $b\in A\cap B$ (because $A=A\cap B$) which means that $b\in B$


*Therefore, if $b\in A\cup B$, then $b\in B$, which means that $A\cup B\subseteq B$

*We also know that (always) $B\subseteq A\cup B$

*We conclude that $B=A\cup B$

