Equivalence of Definitions of Principal Ideal

Theorem
Let $\struct {S, \preceq}$ be a preordered set.

Let $I$ be an ideal in $S$.

Definition $1$ implies Definition $2$
Assume that
 * $\exists x \in I: x$ is upper bound for $I$

We will prove that
 * $I \subseteq x^\preceq$

Let $y \in I$.

By definition of upper bound:
 * $y \preceq x$

Thus by definition of lower closure of element:
 * $y \in x^\preceq$

We will prove that
 * $x^\preceq \subseteq I$

Let $y \in x^\preceq$.

By definition of lower closure of element:
 * $y \preceq x$

Thus by definition of lower section:
 * $y \in I$

Thus by definition of set equality:
 * $\exists x \in S: I = x^\preceq$

Definition $2$ implies Definition $1$
Assume that:
 * $\exists x \in S: I = x^\preceq$

By definition of reflexivity:
 * $x \preceq x$

Thus by definition of lower closure of element:
 * $x \in I$

Let $y \in I$.

Thus by definition of lower closure of element:
 * $y \preceq x$