Prove that:

where x is a set and $\displaystyle Px$ is the power set of x.$\displaystyle \bigcup Px=x \subset P \bigcup x= P \bigcup P \bigcup x$

This is the last part of the question, the former parts required me to prove:

$\displaystyle x \subset y \Rightarrow \bigcup x \subset \bigcup y$

$\displaystyle x \subset y \Leftrightarrow Px \subset Py$

$\displaystyle x \subset Py \Leftrightarrow \bigcup x \subset y $

where x and y are sets.

I can't get anywhere with this! I've tried using $\displaystyle x \subset Px$ but I can't get it to work out. My main problem is that the first three parts of the questions gave me something to work with (ie. condition 1 allows me to use $\displaystyle x \subset y$). The last part doesn't give me any properties to use so i'm having some trouble trying to use the things I already know.

Does anyone have any ideas?