The following definition is a generalization of the concept of minimal elements in a set and of a well-order.
Let $V$ be a set, and let $R\subseteq V\times V$ be a binary relation. The relation $R$ is called well-founded if every non-empty subset $S\subseteq V$ contains a minimal element $m$ with respect to $R$. In other words, no element $x\in S$ is in the left-sided relation $xRm.$ Formally:
$$\forall S\subseteq V\; \exists x\in S \Rightarrow \exists m\in S \quad \forall x\in S \quad (x,m)\not\in R.$$
Examples: 2 3 4
Motivations: 6 7
Proofs: 8 9 10