Theorem wfaxsep

Description: The class of well-founded sets models the Axiom of Separation ax-sep . Actually, our statement is stronger, since it is an instance of Separation only when all quantifiers in ph are relativized to W . Part of Corollary II.2.5 of Kunen2 p. 112. (Contributed by Eric Schmidt, 29-Sep-2025)

	Ref	Expression
Hypothesis	wfax.1	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]$$
Assertion	wfaxsep	$$⊢\forall z\in W\exists y\in W\forall x\in W\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wfax.1	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]$$
2		ssclaxsep	$$⊢\forall z\in W𝒫z\subseteq W\to \forall z\in W\exists y\in W\forall x\in W\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)$$
3		pwwf	$$⊢z\in \bigcup R_1\left[\mathrm{On}\right]\leftrightarrow 𝒫z\in \bigcup R_1\left[\mathrm{On}\right]$$
4		r1elssi	$$⊢𝒫z\in \bigcup R_1\left[\mathrm{On}\right]\to 𝒫z\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
5	3 4	sylbi	$$⊢z\in \bigcup R_1\left[\mathrm{On}\right]\to 𝒫z\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
6	1	eleq2i	$$⊢z\in W\leftrightarrow z\in \bigcup R_1\left[\mathrm{On}\right]$$
7	1	sseq2i	$$⊢𝒫z\subseteq W\leftrightarrow 𝒫z\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
8	5 6 7	3imtr4i	$$⊢z\in W\to 𝒫z\subseteq W$$
9	2 8	mprg	$$⊢\forall z\in W\exists y\in W\forall x\in W\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)$$