Theorem wfaxext

Description: The class of well-founded sets models the axiom of Extensionality ax-ext . Part of Corollaray II.2.5 of Kunen2 p. 112. (Contributed by Eric Schmidt, 11-Sep-2025)

	Ref	Expression
Assertion	wfaxext	⊢ ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∀ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ( ∀ 𝑧 ∈ ∪ ( 𝑅1 “ On ) ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		trwf	⊢ Tr ∪ ( 𝑅1 “ On )
2		traxext	⊢ ( Tr ∪ ( 𝑅1 “ On ) → ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∀ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ( ∀ 𝑧 ∈ ∪ ( 𝑅1 “ On ) ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 ) )
3	1 2	ax-mp	⊢ ∀ 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∀ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ( ∀ 𝑧 ∈ ∪ ( 𝑅1 “ On ) ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )