= Foundation in the cumulative hierarchy without ambient Foundation
{c}
The <cumulative hierarchy> is a transitive class even without the ambient <Axiom of foundation>. If $a$ is a nonempty member, choose the least <ordinal> $\alpha$ with $a\cap V_\alpha\ne\varnothing$. It is a successor $\beta+1$. An element $x$ of that intersection is a subset of $V_\beta$, while $a$ has no element in $V_\beta$, so $a\cap x=\varnothing$. This proves internal <Foundation>. If the hierarchy also satisfies the other <ZF> axioms, restricting every <quantifier> of a finite <ZF> proof to this definable class translates it into a proof from <ZF without foundation>. Thus adjoining <Foundation> preserves consistency relative to the remaining axioms.
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