正則性公理
∀ x ∈ W F ( rank  ( x ) < α ⟺ ∃ β < α ; x ∈ R ( β + 1 ) ⟺ x ∈ R ( α ) ) {\displaystyle \forall x\in {\mathit {WF}}{\bigl (}\operatorname {rank} (x)<\alpha \iff \exists \beta <\alpha ;x\in R(\beta +1)\iff x\in R(\alpha ){\bigr )}}
ランクを計算するときに次の補題を使う。
y ∈ W F {\displaystyle y\in {\mathit {WF}}} のとき、
x ∈ y ⟹ x ∈ W F {\displaystyle x\in y\implies x\in {\mathit {WF}}}
かつ
rank  ( x ) < rank  ( y ) {\displaystyle \operatorname {rank} (x)<\operatorname {rank} (y)}
rank  ( y ) = α {\displaystyle \operatorname {rank} (y)=\alpha } とすると y ∈ R ( α + 1 ) = P ( R ( α ) ) {\displaystyle y\in R(\alpha +1)={\mathcal {P}}(R(\alpha ))}
x ∈ y {\displaystyle x\in y} ならば x ∈ R ( α ) = { x ∈ W F ∣ rank  ( x ) < α } {\displaystyle x\in R(\alpha )=\{x\in {\mathit {WF}}\mid \operatorname {rank} (x)<\alpha \}} だから rank  ( x ) < α {\displaystyle \operatorname {rank} (x)<\alpha }