= Transitivity and strict growth of the cumulative hierarchy
{title2=$V_\alpha\subsetneq V_{\alpha+1}$}
Each stage of the <cumulative hierarchy> is a <transitive set>, and the stages increase strictly. If $A$ is transitive, then $A\subseteq\mathcal P(A)$ and $\mathcal P(A)$ is transitive: $x\in y\subseteq A$ gives $x\in A$ and then $x\subseteq A$. Unions of increasing <transitive sets> are transitive. These observations prove the assertions by <transfinite induction>, with strictness witnessed by $V_\alpha\in V_{\alpha+1}$ and $V_\alpha\notin V_\alpha$ from the <Axiom of foundation>.
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