Metamath Proof Explorer

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	$⊢ W = ⋃ R_{1} [On]$
	Assertion	wfaxreg	$⊢ \forall x \in W (\exists y \in W y \in x \to \exists y \in W (y \in x \land \forall z \in W (z \in y \to \neg z \in x)))$

Proof

Step	Hyp	Ref	Expression
1		wfax.1	$⊢ W = ⋃ R_{1} [On]$
2	1	eqimssi	$⊢ W \subseteq ⋃ R_{1} [On]$
3		sswfaxreg	$⊢ W \subseteq ⋃ R_{1} [On] \to \forall x \in W (\exists y \in W y \in x \to \exists y \in W (y \in x \land \forall z \in W (z \in y \to \neg z \in x)))$
4	2 3	ax-mp	$⊢ \forall x \in W (\exists y \in W y \in x \to \exists y \in W (y \in x \land \forall z \in W (z \in y \to \neg z \in x)))$