Intersection of Ordinals is Smallest

Theorem
Let $A$ be a non-empty set or class of ordinals.

Let $m = \bigcap A$ be the intersection of all the elements of $A$.

Then $m$ is the smallest element of $A$.

Proof
By Intersection of Ordinals is Ordinal, $m$ is an ordinal.

By Intersection Largest, $m \subseteq a$ for each $a \in A$.

It remains to show that $m \in A$.

Let $m^+ = m \cup \left\{{m}\right\}$ be the successor of $m$.

By Relation between Two Ordinals, for each $a \in A$, either $m^+ \subseteq a$ or $a \in m^+$.

If $m^+ \subseteq a$ for all $a \in A$, then by Intersection Largest $m^+ \subseteq m$, contradicting the fact that Ordinal is not Element of Itself.

Thus there is an $a \in A$ such that $a \in m^+$.

Thus $a \in m$ or $a = m$.

If $a \in m$, then $a \subsetneqq m$, contradicting the fact that $m \subseteq a$.

Thus $a = m$, so $m \in A$.