= Intersection-power-set predecessor relation
{title2=$xRy\iff\mathcal P(x\cap y)\subseteq y$}
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 $d\in d\iff d\notin d$.
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