Transitive Set is Proper Subset of Ordinal iff Element of Ordinal/Corollary

Corollary to Transitive Set is Proper Subset of Ordinal iff Element of Ordinal
Let $A$ and $B$ be an ordinals.

Then:
 * $A \subsetneq B \iff A \in B$

Proof
We have that Ordinal is Transitive.

The result follows directly from Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

Source

 * : $\S 7.8$