Directed iff Finite Subsets have Upper Bounds

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

Let $H$ be a non-empty subset of $S$.

Then $H$ is directed


 * for every a finite subset $A$ of $H$
 * $\exists h \in H: \forall a \in A: a \precsim h$

Sufficient Condition
Let $R$ be directed.

We will prove by induction of the cardinality of finite subset of $H$.

Base case

 * $\forall A \subseteq H: \left\vert{A}\right\vert = 0 \implies \exists h \in H: \forall a \in A: a \precsim h$

where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.

Let $A \subseteq H$ such that
 * $\left\vert{A}\right\vert = 0$

By Cardinality of Empty Set:
 * $A = \varnothing$

By definition of empty set:
 * $\exists h: h \in H$

Thus by definition of empty set:
 * $\exists h \in H: \forall a \in A: a \precsim h$

Induction Hypothesis

 * $\forall A \subseteq H: \left\vert{A}\right\vert = n \implies \exists h \in H: \forall a \in A: a \precsim h$

Induction Step

 * $\forall A \subseteq H: \left\vert{A}\right\vert = n+1 \implies \exists h \in H: \forall a \in A: a \precsim h$

Let $A \subseteq H$ such that
 * $\left\vert{A}\right\vert = n+1$

By definition of cardinality of finite set:
 * $A \sim \N_{< n+1}$

where $\dim$ denotes set equivalence.

By definition of set equivalence:
 * there exists a bijection $f: \N_{< n+1} \to A$

By Restriction of Injection is Injection:
 * $f \restriction_{\N_{< n}}: \N_{< n} \to f^\to\left({\N_{< n}}\right)$ is an injection.

By definition
 * $f \restriction_{\N_{< n}}: \N_{< n} \to f^\to\left({\N_{< n}}\right)$ is a surjection.

By definition
 * $f \restriction_{\N_{< n}}: \N_{< n} \to f^\to\left({\N_{< n}}\right)$ is a bijection.

By definition of set equivalence:
 * $\N_{< n} \sim f^\to\left({\N_{< n}}\right)$

By definition of cardinality of finite set:
 * $\left\vert{f^\to\left({\N_{< n}}\right)}\right\vert = n$

By definitions of image of set and subset:
 * $f^\to\left({\N_{< n}}\right) \subseteq A$

By Subset Relation is Transitive:
 * $f^\to\left({\N_{< n}}\right) \subseteq H$

By Induction Hypothesis:
 * $\exists h \in H: \forall a \in f^\to\left({\N_{< n}}\right): a \precsim h$

By definition $\N_{< n+1}$
 * $n \in \N_{< n+1}$

By definition of mapping:
 * $f \left({n}\right) \in A$

By definition of subset:
 * $f \left({n}\right) \in H$

By definition of directed subset:
 * $\exists h' \in H: f \left({n}\right) \precsim h' \land h \precsim h'$

Let $a \in A$.

Then by definitions of union and singleton:
 * $a \in f^\to\left({\N_{< n} }\right) \lor a = f\left({n }\right)$

So:
 * $a \precsim h \lor a = f\left({n }\right)$

Thus by definition of transitivity:
 * $a \precsim h'$

Thus:
 * $\exists h' \in H: \forall a \in A: a \precsim h'$

Necessary Condition
Assume that
 * for every a finite subset $A$ of $H$
 * $\exists h \in H: \forall a \in A: a \precsim h$

Let $a, b \in H$

By definition of subset:
 * $\left\{ {a, b}\right\} \subseteq H$
 * $\left\{ {a, b}\right\}$ is finite

By assumption:
 * $\exists h \in H: \forall c \in \left\{ {a, b}\right\}: c \precsim h$

Thus by definition of unordered tuple:
 * $\exists h \in H: a \precsim h \land b \precsim h$

Thus by definition
 * $H$ is directed.