Why is $\mathcal{P}(\bigcup A)\in A$ equivalent to $\mathcal{P}(\bigcup A)\subseteq\bigcup A$?

Question

$\mathcal{P}(\bigcup A)\in A\leftrightarrow\mathcal{P}(\bigcup A)\subseteq\bigcup A$

I know this brings out a contradiction but it is part of a bigger proof and I would like to know why these two statements are equivalent?

Answer 1

$\cup A\in\wp(\cup A)$ so $\wp(\cup A)\in A$ would imply that $\cup A\in\cup A$.

Also the right side leads to $\cup A\in\cup A$ because $\cup A\in\wp(\cup A)$.

If the axiom of regularity is accepted then this is not true.

If $p$ and $q$ are both not true then $p\iff q$ is true.