Collection of All Ordered Sets is not Set

Theorem
Let $\mathrm {OS}$ denote the collection of all ordered sets.

Then $\mathrm {OS}$ is not a set.

Proof
Let $C$ be the collection of all singletons:
 * $\set {x: \exists y: x = \set y}$.

Define a mapping $\map f {\set y} = \RR$ where $\RR$ is a reflexive relation on $\set y$.

By Reflexive Relation on Singleton is Well-Ordering, $\RR$ is an ordering.

Thus:
 * $f:C \to \mathrm {OS}$

By Equality of Ordered Pairs, if $\map f {y_1} = \map f {y_2}$ then $y_1 = y_2$.

Thus $f$ is an injection.

By definition of cardinality, $C$ is the collection of all sets which are equivalent to $\set \O$.

By Collection of Sets Equivalent to Set Containing Empty Set is Proper Class, $C$ is proper.

Thus, by Injection from Proper Class to Class, $\mathrm {OS}$ is proper.