Theorem axtco1 36838

Description: Strong form of the Axiom of Transitive Containment. See ax-tco 36837 for more information. In particular, this theorem generalizes the statement of ax-tco 36837, 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.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑦,𝑤,𝑧

Proof of Theorem axtco1

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2151	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
2	1	anbi1d 640	. . 3 ⊢ (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 1943	. 2 ⊢ (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 36837	. 2 ⊢ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
5	3, 4	chvarvv 2011	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1560  ∃wex 1801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-tco 36837

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802

This theorem is referenced by:  axtco2  36839  axtco1g  36841