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A standard model of set theory interprets membership by the ambient membership relation restricted to its domain. A transitive model additionally has a transitive set or class as its domain. Standardness alone does not impose transitivity.

The transitive closure is obtained by $x_0=x$, $x_{n+1}=\bigcup x_n$, $t={\bigcup }_{n<\omega }{x}_n$. This is transitive, contains $x$ as a subset, and is contained in any other transitive set containing $x$, by induction. Its graph has a Sigma-one formula in set theory saying there exist $\omega$, a sequence with this recursion, and its range. The bounded definition of omega, function and union predicates keep the matrix bounded. Existence and uniqueness in ZF express failure of $y=\mathrm{TC}(x)$ by the existential assertion that the construction produces some $z\ne y$. Thus the graph is a Delta-one formula modulo ZF, and Delta-one absoluteness makes transitive closure absolute between nested transitive models of ZF.

Using the convention that $\mathrm{TC}(x)$ is the least transitive set containing $x$ as a subset, put $x_0=x$, $x_{n+1}=\bigcup x_n$ and $\mathrm{TC}(x)={\bigcup }_{n<\omega }{x}_n$. If $\mathrm{rank}x=\alpha$, every element encountered is an iterated member of an original element and has rank below $\alpha$. Thus the closure has rank at most $\alpha$, while $x\subseteq \mathrm{TC}(x)$ gives the reverse inequality. A convention requiring $x$ itself to be an element instead has rank $\alpha +1$.

If $X$ has a well-founded extensional inherited membership relation and $T\subseteq X$ is a transitive set, its Mostowski collapse theorem map $\pi$ fixes every member of $T$. The recursion $\pi (x)=\{\pi (y):y\in x\cap X\}$ and well-founded relation induction give this pointwise. If also $T\in X$, then $\pi (T)=T$.

A transitive model of set theory is a transitive set or transitive class whose membership relation is the ambient membership relation and which satisfies the specified set-theoretic axioms.

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.