Theorem axtco2 37013

Description: Weak form of the Axiom of Transitive Containment. See ax-tco 37011 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.)

Assertion

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

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

Proof of Theorem axtco2

Step	Hyp	Ref	Expression
1		axtco1 37012	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
2		elequ1 2149	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
3	2	biimprcd 253	. . . . . 6 ⊢ (𝑥 ∈ 𝑦 → (𝑧 = 𝑥 → 𝑧 ∈ 𝑦))
4	3	imim1d 83	. . . . 5 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑧 = 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))))
5	4	alimdv 1945	. . . 4 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → ∀𝑧(𝑧 = 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))))
6		jao 974	. . . . 5 ⊢ ((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 1844	. . . 4 ⊢ (∀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 40	. . 3 ⊢ (𝑥 ∈ 𝑦 → (∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → ∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))))
9	8	imp 411	. 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))) → ∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
10	1, 9	eximii 1866	1 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∨ wo 860  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-tco 37011

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809

This theorem is used by:  axtco1from2  37014  axtcond  37017