Rank of a union of a set

Question

I'm exploring the Von Neumann Universe and specifically trying to determine rank($\cup x$) in terms of rank($x$). The answer seems to be depending on whether rank($x$) is a successor, or a limit ordinal. I've already shown that for limit ordinals $\alpha$, we have $\cup \alpha = \alpha$, so their ranks coincide as well. The proof of the case where $\alpha$ is a successor ordinal however, eludes me.

Answer 1

$$\text{rank}(\cup x)=\sup\{\text{rank}(y)+1:y\in\cup x\}$$ $$\leq\sup\{\text{rank}(y):y\in x\}\leq \text{rank}(\cup x)$$

Therefore $\text{rank}(\cup x)=\sup\{\text{rank}(y):y\in x\}$. If the $\sup$ is attained then $$\text{rank}(\cup x)+1=\text{rank}( x)$$ and if the $\sup$ is not attained then $$\text{rank}(\cup x)=\text{rank}( x)$$