Theorem axtco1from2 36663

Description: Strong form axtco1 36661 of the Axiom of Transitive Containment, derived from the weak form axtco2 36662. See ax-tco 36660 for more information. As written, the proof uses ax-pr 5368 via el 5383, but we could alternatively use ax-pow 5300 via elALT2 5304. Use axtco1 36661 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 2121	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
2	1	anbi1d 632	. . 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 1923	. 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 36662	. . . 4 ⊢ ∃𝑦∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
5		orc 868	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦))
6		elequ2 2129	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑢 → (𝑣 ∈ 𝑧 ↔ 𝑣 ∈ 𝑢))
7	6	biimprd 248	. . . . . . . . . 10 ⊢ (𝑧 = 𝑢 → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑧))
8		elequ1 2121	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑧 ↔ 𝑣 ∈ 𝑧))
9		elequ1 2121	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑦 ↔ 𝑣 ∈ 𝑦))
10	8, 9	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑣 → ((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) ↔ (𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑦)))
11	10	spvv 1990	. . . . . . . . . 10 ⊢ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑦))
12	7, 11	syl9 77	. . . . . . . . 9 ⊢ (𝑧 = 𝑢 → (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑦)))
13	5, 12	embantd 59	. . . . . . . 8 ⊢ (𝑧 = 𝑢 → (((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑦)))
14	13	spimvw 1988	. . . . . . 7 ⊢ (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑣 ∈ 𝑢 → 𝑣 ∈ 𝑦))
15	14	com12 32	. . . . . 6 ⊢ (𝑣 ∈ 𝑢 → (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → 𝑣 ∈ 𝑦))
16		olc 869	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦))
17	16	imim1i 63	. . . . . . 7 ⊢ (((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → (𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
18	17	alimi 1813	. . . . . 6 ⊢ (∀𝑧((𝑧 = 𝑢 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)) → ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
19	15, 18	jca2 513	. . . . 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 1919	. . . 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 20	. . 3 ⊢ (𝑣 ∈ 𝑢 → ∃𝑦(𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))))
22		el 5383	. . 3 ⊢ ∃𝑢 𝑣 ∈ 𝑢
23	21, 22	exlimiiv 1933	. 2 ⊢ ∃𝑦(𝑣 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
24	3, 23	chvarvv 1991	1 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-pr 5368  ax-tco 36660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782

This theorem is referenced by: (None)