I have seen all the available answers to this question here and I am still in doubt. I know that any set x containing a single element, ie itself, can easily shown to be non-existent.
But, how do we go about proving that the following is not possible? -
x = {y, x} - were y is another entity completely disjoint from x. Say, y = 1.
If $t\in t$, then $\{t\}\ne \emptyset$ and $t\in \{t\}\cap t$, therefore $\{t\} $ fails the axiom of regularity. Specialise this argument to the case $t=\{x,y\}$, and you obtain that $x=\{x,y\}$ implies that $\{x\}$ (which is also equal to $\{\{x,y\}\}$) fails regularity.