Given any sets $X$ and $Y$ with $(X\times X)\cap Y\subseteq X$ must it be true that $(X\times X)\cap Y=\emptyset$?
I believe it is, I mean $(X\times X)\cap Y\subseteq X\times X$ so if its not empty $(X\times X)\cap Y$ then its a set of ordered pairs of the elements of $X$ contained in $X$ which doesn't seem right to me.
However I've made dumb/trivial mistakes with the axiom of regularity before and other times when I'm dealing with multiple sets operating on each other while also being defined in terms of each other and with my naive intuition often causing me to think of elements as objects in their own right rather then just sets themselves belonging inside other sets this sometimes can cause me to make errors.
Anyway to iterate if this is trivial/obvious I am sorry. I thought I had an okay grasp on set theory but as I've been playing around with various definitions it seems I know far, far less then I thought I did. So I'm often doubting myself when manipulating objects containing sets within sets etc. or where I have a set that is sort of related to another set it contains in such a manner that it seems like something is self referencing from within itself to another set lower down contained in it.