How do I make sense of $\{ n \} \cap n$? I've been learning set theory, and I've come across an exercise in which I'm trying to prove that $\forall x \forall y x \in y \rightarrow y \neq x$. I want to use the axiom of foundation to prove this, but I'm stuck making sense of that axiom for the base case in which a set contains something like a single integer.
If I have $x=\{ y, 5 \}$ for example, and $y=\{5\}$, the axiom of foundation seems to require that $x\cap5=\emptyset$, but it seems like this set should intersect 5. How do I make sense of this? Is the intersection of a set and an integer just always defined as $\emptyset$? Is it even possible to define the intersection of the set and a non-set member of the set?

In axiomatic set theory, everything is a set. In particular the integers. Which sets are the integers may vary, but the standard is to use von Neumann's definition:


*

*$0=\varnothing$, and 

*$n+1=n\cup\{n\}$.


So for example $3=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}=\{0,1,2\}$.
And you don't really need to use a "base" case. Just show that if $x\in y$ and $x=y$ then you can prove the negation of the axiom of foundation.
