Upper Closure of Singleton

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Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $s$ be an element of $S$.

Then:

$\set s^\succeq = s^\succeq$

where:

$\set s^\succeq$ denotes the upper closure of $\set s$
$s^\succeq$ denotes the upper closure of $s$


Proof

\(\ds \set s^\succeq\) \(=\) \(\ds \bigcup \set {t^\succeq: t \in \set s}\) Definition of Upper Closure of Subset
\(\ds \) \(=\) \(\ds \bigcup \set {s^\succeq}\) Definition of Singleton
\(\ds \) \(=\) \(\ds s^\succeq\) Union of Singleton

$\blacksquare$


Sources