Theorem wfaxext

Description: The class of well-founded sets models the axiom of Extensionality ax-ext . Part of Corollary II.2.5 of Kunen2 p. 112. (Contributed by Eric Schmidt, 11-Sep-2025) (Revised by Eric Schmidt, 29-Sep-2025)

	Ref	Expression
Hypothesis	wfax.1	⊢ 𝑊 = ∪ ( 𝑅1 “ On )
Assertion	wfaxext	⊢ ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 ( ∀ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		wfax.1	⊢ 𝑊 = ∪ ( 𝑅1 “ On )
2		trwf	⊢ Tr ∪ ( 𝑅1 “ On )
3		treq	⊢ ( 𝑊 = ∪ ( 𝑅1 “ On ) → ( Tr 𝑊 ↔ Tr ∪ ( 𝑅1 “ On ) ) )
4	1 3	ax-mp	⊢ ( Tr 𝑊 ↔ Tr ∪ ( 𝑅1 “ On ) )
5	2 4	mpbir	⊢ Tr 𝑊
6		traxext	⊢ ( Tr 𝑊 → ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 ( ∀ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 ) )
7	5 6	ax-mp	⊢ ∀ 𝑥 ∈ 𝑊 ∀ 𝑦 ∈ 𝑊 ( ∀ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦 ) → 𝑥 = 𝑦 )