Equivalence of Definitions of Principal Ideal

Theorem
Let $\left({S, \preceq}\right)$ be a preordered set.

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

Then
 * the definitions of principal ideal are equivalent,

That means that
 * $\exists x \in I: x$ is upper bound for $I$


 * $\exists x \in S: I = x^\preceq$

where $x^\preceq$ denotes the lower closure of $x$.

Sufficient Condition
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 set:
 * $y \in I$

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

Necessary Condition
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:
 * $y \preceq x$