I'm not too sure about this one either...
Show that P(X) is a subset of X is false for any X. In particular, P(X) does not equal X for any X.
Construct the set . Notice that therefore . However, , to show this, assume to contrary that . Now what we have is essentially Russel's paradox, because if we must have either or . If then by construction , a contradiction. If then by construction for and , a contradiction. Thus, we must have .