This must be trivial, so my apologies to specialists in logic. I am trying to study the Morse-Kelley theory, and this is a continuation of my previous question here.
Suppose $X$ is a class, that does not coincide with the class $V$ of all sets: $$ X\ne V. $$ Is it possible that this automatically means that there exists a subset $Y\subseteq X$ which is not an element of $X$?
Yes, this is correct. By the axiom of foundation applied to $V\setminus X$, if $X$ is not all of $V$, then there is some $Y\in V\setminus X$ such that no element of $Y$ is in $V\setminus X$. This is exactly a set $Y$ such that $Y\subseteq X$ but $Y\not\in X$.
(Incidentally, if you want to do the same argument in ZF, you have to do something more complicated like take a $Y\not\in X$ and then apply foundation to the transitive closure of $Y$, since you can't apply foundation to classes directly. Or you could use Scott's trick, but I'm guessing that would be circular in this context since this statement is likely being used to prove that $V=\bigcup V_\alpha$.)
