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 = ∪ ( 𝑅1 “ On )
Assertion	wfaxreg	⊢ ∀ 𝑥 ∈ 𝑊 ( ∃ 𝑦 ∈ 𝑊 𝑦 ∈ 𝑥 → ∃ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		wfax.1	⊢ 𝑊 = ∪ ( 𝑅1 “ On )
2	1	eqimssi	⊢ 𝑊 ⊆ ∪ ( 𝑅1 “ On )
3		sswfaxreg	⊢ ( 𝑊 ⊆ ∪ ( 𝑅1 “ On ) → ∀ 𝑥 ∈ 𝑊 ( ∃ 𝑦 ∈ 𝑊 𝑦 ∈ 𝑥 → ∃ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) ) )
4	2 3	ax-mp	⊢ ∀ 𝑥 ∈ 𝑊 ( ∃ 𝑦 ∈ 𝑊 𝑦 ∈ 𝑥 → ∃ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥 ) ) )