I suspect the solution lies in the phrase "in fact empty". The empty relation is trivially well founded. So look at why set membership for y should be the empty

I suspect the solution lies in the phrase "in fact empty". The empty relation is trivially well founded. So look at why set membership for y should be the empty relation. I'll take a shot (with the caveat that I'm no logician, and I'm not entirely convinced by this argument myself). It might be helpful first to remove x from the picture and define y as: y = {{y}} and y \neq {y} First note that there can be at most one distinct member of {{y}}, which is {y}. So if there exists z member of y, then z = {y}. Then y is a member of z, and by transitivity, y is member of y, and so y = z = {y}. This contradicts the assumption that y \neq {y}, so there can be no such z.