= Power-set-predecessor relation
{title2=$xEy\iff\mathcal P(x)\subseteq y$}
This <well-founded relation> is well-founded without the <axiom of choice> or <Axiom of foundation>. If a nonempty set $A$ lacked a minimal member, put $b=\bigcap A$. For each $y\in A$, some $x\in A$ has $\mathcal P(x)\subseteq y$, and $b\subseteq x$ gives $\mathcal P(b)\subseteq y$. Hence $\mathcal P(b)\subseteq b$. The <Cantor diagonal argument> applied to $d=\{u\in b:u\notin u\}$ contradicts this inclusion.
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