Past exam of the logic and set theory course of the University of Oxford / 2011 / B / 5 / a / i / Solution

A transitive set $X$ is one for which every member is a subset: $\forall y\in X(y\subseteq X)$, equivalently $\bigcup X\subseteq X$. If $x\in y\in X$, the subset condition puts $x\in X$. Conversely, if this membership implication always holds, every element of each $y\in X$ belongs to $X$, so $y\subseteq X$. Thus