Prove that "if $A\cap B=A\cap C$ and $A\cup B=A \cup C$, then $B = C$" by contrapositive. I've been trying to prove this for the last $30$ minutes or so, but my proof made me very confused.
I assumed that $B\neq C$, and I have to prove that $A \cap B \neq A \cap C$ or $A \cup B \neq A \cup C$.
Since $B\neq C$ we have two cases:

*

*$x\in B$ and $x\notin C$

*$x\in C$ and $x \notin B$
For case $1$ we have again two subcases

*

*$x\in A$

*$x \notin A$
For subcase $1,$ $x\in A$, we have that $x \in A \cap B$ and $x\notin A \cap C$, which is the desired result.
For subcase $2$ instead I get super confused,  what happens if $x \notin A$? I can't say anything right? Because maybe $A \cap B$ is empty but $A \cap C$ is not, since we just said that $x\in B$, $x \notin A$, and $x \notin C$  without saying anything about weather $A$ and $C$ have any other elements, right? This just means that i can't proceed with the proof anymore and I'm stuck and I have to change "strategy" right?
Please don't give me the solution of the exercises since I want to do it by myself. In case I can't I will ask.

Since $x\notin A$, you can't say anything about $A\cap B$ and $A\cap C$, but what happens with $A\cup B$ and $A\cup C$?
