strict well-order

A strict well-order is a homogeneous relation that is irreflexive, transitive, antisymmetric, connected, and has a least element in every subset.

Formally,
a binary relation $<$ over a set $S$ is a strict well-order if for all $a,b,c$ in $S$,

• $a\nless a$ (irreflexive)
• If $a<b$ and $b<c$ then $a<c$ (transitive)
• If $a<b$ then $b\nless a$ (antisymmetric)
• $a<b$ or $b<a$ (connected)
• $\mathrm{\exists }!l_W\in W$ such that $l_W<w$ for all $w\ne l_W$ in every subset $W\subseteq X$ (least element)

A strict well-order is a strict total order such that every subset has a least element.