# Thread: More Axiom of Union and Powerset

1. ## More Axiom of Union and Powerset

Proof.

2. Let $y \in x$ and let $z \in y$. Then $z \in \cup x$ (definition of union). So $y \subseteq \cup x$. It follows that $y \in \mathcal{P} \left(\cup x \right)$ (definition of power set).
We showed that $y \in x$ implies $y \in \mathcal{P} \left(\cup x \right)$, so $x \subseteq \mathcal{P} \left(\cup x \right)$.

Example of strict inclusion: $x = \{\{a\}\}$. We have $\cup x = \{a\}$, $\mathcal{P}\left(\cup x \right) = \{\emptyset, \{a\}\}$. We see that $x = \{\{a\}\} \subsetneq \{\emptyset, \{a\}\} = \mathcal{P}\left(\cup x \right)$.