Characterization of Minimal Element

Theorem
Let $C$ be a class.

Let $\prec$ be a Definition:relation on $C$.

Let $B$ be a subclass of $C$.

Let $x \in B$.

Let $S_x = \left\{{y \in C \mid y \prec x \text{ and } y \ne x}\right\}$ be the initial segment of $x$ in $C$.

Then $x$ is a minimal element of $B$ iff $B \cap S_x = \varnothing$.

Implication
Suppose $x$ is a minimal element of $B$.

Then for each $z \in B$ such that $z \ne x$, $z \not\prec x$.

Thus $S_x \cap B = \varnothing$.

Reverse implication
Suppose that $B \cap S_x = \varnothing$. Then