Theorem axtco1from2 37263

Description: Strong form axtco1 37261 of the Axiom of Transitive Containment, derived from the weak form axtco2 37262. See ax-tco 37260 for more information. As written, the proof uses ax-pr 5390 via el 5405, but we could alternatively use ax-pow 5326 via elALT2 5330. Use axtco1 37261 instead. (Contributed by Matthew House, 6-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axtco1from2	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

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

Proof of Theorem axtco1from2

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2151	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
2	1	anbi1d 643	. . 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 1953	. 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		axtco2 37262	. . . 4 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
5		orc 881	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦))
6		elequ2 2159	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑣 ∈ 𝑧 ↔ 𝑣 ∈ 𝑢))
7	6	biimprd 251	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑧))
8		elequ1 2151	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
9		elequ1 2151	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑦 ↔ 𝑣 ∈ 𝑦))
10	8, 9	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) ↔ (𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑦)))
11	10	spvv 2020	. . . . . . . . . 10 ⊢ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑦))
12	7, 11	syl9 78	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑦)))
13	5, 12	embantd 60	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑦)))
14	13	spimvw 2018	. . . . . . 7 ⊢ (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑦))
15	14	com12 33	. . . . . 6 ⊢ (𝑣 ∈ 𝑢 → (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → 𝑣 ∈ 𝑦))
16		olc 882	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦))
17	16	imim1i 64	. . . . . . 7 ⊢ (((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
18	17	alimi 1843	. . . . . 6 ⊢ (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
19	15, 18	jca2 523	. . . . 5 ⊢ (𝑣 ∈ 𝑢 → (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))))
20	19	eximdv 1949	. . . 4 ⊢ (𝑣 ∈ 𝑢 → (∃𝑦∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → ∃𝑦(𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))))
21	4, 20	mpi 21	. . 3 ⊢ (𝑣 ∈ 𝑢 → ∃𝑦(𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))))
22		el 5405	. . 3 ⊢ ∃𝑢 𝑣 ∈ 𝑢
23	21, 22	exlimiiv 1963	. 2 ⊢ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
24	3, 23	chvarvv 2021	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-pr 5390  ax-tco 37260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812

This theorem is used by: (None)