Theorem wfaxinf2

Description: The class of well-founded sets models the Axiom of Infinity ax-inf2 . Part of Corollary II.2.12 of Kunen2 p. 114. (Contributed by Eric Schmidt, 19-Oct-2025)

	Ref	Expression
Hypothesis	wfax.1	⊢ 𝑊 = ∪ ( 𝑅1 “ On )
Assertion	wfaxinf2	⊢ ∃ 𝑥 ∈ 𝑊 ( ∃ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑊 ¬ 𝑧 ∈ 𝑦 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑥 ∧ ∀ 𝑤 ∈ 𝑊 ( 𝑤 ∈ 𝑧 ↔ ( 𝑤 ∈ 𝑦 ∨ 𝑤 = 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		onwf	⊢ On ⊆ ∪ ( 𝑅1 “ On )
7		omelon	⊢ ω ∈ On
8	6 7	sselii	⊢ ω ∈ ∪ ( 𝑅1 “ On )
9	8 1	eleqtrri	⊢ ω ∈ 𝑊
10		omelaxinf2	⊢ ( ( Tr 𝑊 ∧ ω ∈ 𝑊 ) → ∃ 𝑥 ∈ 𝑊 ( ∃ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑊 ¬ 𝑧 ∈ 𝑦 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑥 ∧ ∀ 𝑤 ∈ 𝑊 ( 𝑤 ∈ 𝑧 ↔ ( 𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦 ) ) ) ) ) )
11	5 9 10	mp2an	⊢ ∃ 𝑥 ∈ 𝑊 ( ∃ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑧 ∈ 𝑊 ¬ 𝑧 ∈ 𝑦 ) ∧ ∀ 𝑦 ∈ 𝑊 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ∈ 𝑊 ( 𝑧 ∈ 𝑥 ∧ ∀ 𝑤 ∈ 𝑊 ( 𝑤 ∈ 𝑧 ↔ ( 𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦 ) ) ) ) )