Theorem axtco2

Description: Weak form of the Axiom of Transitive Containment. See ax-tco for more information. In particular, this theorem shows the derivation of the weak form from the strong form. (Contributed by Matthew House, 6-Apr-2026)

	Ref	Expression
Assertion	axtco2	⊢ ∃ 𝑦 ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		axtco1	⊢ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) )
2		elequ1	⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) )
3	2	biimprcd	⊢ ( 𝑥 ∈ 𝑦 → ( 𝑧 = 𝑥 → 𝑧 ∈ 𝑦 ) )
4	3	imim1d	⊢ ( 𝑥 ∈ 𝑦 → ( ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ( 𝑧 = 𝑥 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) )
5	4	alimdv	⊢ ( 𝑥 ∈ 𝑦 → ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ∀ 𝑧 ( 𝑧 = 𝑥 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) )
6		jao	⊢ ( ( 𝑧 = 𝑥 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ( ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ( ( 𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) )
7	6	al2imi	⊢ ( ∀ 𝑧 ( 𝑧 = 𝑥 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) )
8	5 7	syli	⊢ ( 𝑥 ∈ 𝑦 → ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) → ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) )
9	8	imp	⊢ ( ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) → ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) )
10	1 9	eximii	⊢ ∃ 𝑦 ∀ 𝑧 ( ( 𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦 ) → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) )