Ordinal is Transitive/Proof 2

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Every ordinal is a transitive set.


Let $\alpha$ be an ordinal by Definition 2:

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

\((1)\)   $:$   $\alpha$ is a transitive set      
\((2)\)   $:$   the epsilon relation is connected on $\alpha$:    \(\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x \)      
\((3)\)   $:$   $\alpha$ is well-founded.      

Thus $\alpha$ is a priori transitive.