 Intersection-power-set predecessor relation

This relation has well-foundedness without Axiom of foundation or axiom of choice. If a nonempty set $S$ had no minimal member, put $b=\bigcap S$. For each $y\in S$, choose existentially an $x\in S$ with $xRy$. Since $b\subseteq x\cap y$, every subset of $b$ belongs to $y$. Thus $\mathcal{P}(b)\subseteq b$. But $d=\{u\in b:u\notin u\}$ is then an element of $b$, contradicting $d\in d⟺d\notin d$.