Theorem axtco1

Description: Strong form of the Axiom of Transitive Containment. See ax-tco for more information. In particular, this theorem generalizes the statement of ax-tco , allowing it to be written with only three variables, since x need not be distinct from both z and w . (Contributed by Matthew House, 7-Apr-2026)

	Ref	Expression
Assertion	axtco1	⊢ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elequ1	⊢ ( 𝑣 = 𝑥 → ( 𝑣 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) )
2	1	anbi1d	⊢ ( 𝑣 = 𝑥 → ( ( 𝑣 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) ↔ ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) ) )
3	2	exbidv	⊢ ( 𝑣 = 𝑥 → ( ∃ 𝑦 ( 𝑣 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) ) ) )
4		ax-tco	⊢ ∃ 𝑦 ( 𝑣 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) )
5	3 4	chvarvv	⊢ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑤 ( 𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦 ) ) )