Theorem wfaxreg

Description: The class of well-founded sets models the Axiom of Regularity ax-reg . Part of Corollary II.2.5 of Kunen2 p. 112. (Contributed by Eric Schmidt, 19-Oct-2025)

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

Proof

Step	Hyp	Ref	Expression
1		wfax.1	$$⊢W=\bigcup R_1\left[\mathrm{On}\right]$$
2	1	eqimssi	$$⊢W\subseteq \bigcup R_1\left[\mathrm{On}\right]$$
3		sswfaxreg	$$⊢W\subseteq \bigcup R_1\left[\mathrm{On}\right]\to \forall x\in W\left(\exists y\in Wy\in x\to \exists y\in W\left(y\in x\wedge \forall z\in W\left(z\in y\to \neg z\in x\right)\right)\right)$$
4	2 3	ax-mp	$$⊢\forall x\in W\left(\exists y\in Wy\in x\to \exists y\in W\left(y\in x\wedge \forall z\in W\left(z\in y\to \neg z\in x\right)\right)\right)$$