Directed iff Finite Subsets have Upper Bounds

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

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

Then $H$ is directed :


 * for every 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

 * $\forall A \subseteq H: \card A = 0 \implies \exists h \in H: \forall a \in A: a \precsim h$

where $\card A$ denotes the cardinality of $A$.

Let $A \subseteq H$ such that
 * $\card A = 0$

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

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

 * $\forall A \subseteq H: \card A = n \implies \exists h \in H: \forall a \in A: a \precsim h$

Induction Step

 * $\forall A \subseteq H: \card A = n + 1 \implies \exists h \in H: \forall a \in A: a \precsim h$

Let $A \subseteq H$ such that
 * $\card A = n + 1$

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

where $\sim$ denotes set equivalence.

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

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

By definition
 * $f \restriction_{\N_{< n} }: \N_{< n} \to \map {f^\to} {\N_{<n} }$ is a surjection.

By definition
 * $f \restriction_{\N_{< n} }: \N_{< n} \to \map {f^\to} {\N_{<n} }$ is a bijection.

By definition of set equivalence:
 * $\N_{< n} \sim \map {f^\to} {\N_{<n} }$

By definition of cardinality of finite set:
 * $\card {\map {f^\to} {\N_{<n} } } = n$

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

By Subset Relation is Transitive:
 * $\map {f^\to} {\N_{<n} } \subseteq H$

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

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

By definition of mapping:
 * $\map f n \in A$

By definition of subset:
 * $\map f n \in H$

By definition of directed subset:
 * $\exists h' \in H: \map f n \precsim h' \land h \precsim h'$

Let $a \in A$.

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

So:
 * $a \precsim h \lor a = \map f n$

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:
 * $\set {a, b} \subseteq H$
 * $\set {a, b}$ is finite

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

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

Thus by definition:
 * $H$ is directed.