Ordinal is Transitive

Definition
Let $A$ be an Ordinal. Then,


 * $Tr A$

That is - every Ordinal is a transitive class.

Proof
$A$ is an Ordinal iff $\forall x \in A: \{ y \in A : y \subset x \} = x$ by the definition of an ordinal. Thus, $\forall x \in A: \forall y: ( ( y \in A \land y \subset x ) \iff y \in x )$.

The biconditional can be reduced to an implication, and therefore, $\forall x \in A: \forall y \in x: ( y \in A \land y \subset x )$. By simplification of the $\land$ statement, $\forall x \in A: \forall y \in x: y \in A$. By the definition of a subset, $\forall x \in A: x \subseteq A$. Finally, by the definition of a Transitive Class, $Tr A$.