Theorem axuntco 36720

Description: Derivation of ax-un 7681 from ax-tco 36713. Use ax-un 7681 instead. (Contributed by Matthew House, 6-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Proof of Theorem axuntco

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

Step	Hyp	Ref	Expression
1		ax-tco 36713	. 2 ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)))
2		elequ1 2128	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → (𝑣 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
3		elequ2 2136	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑥 → (𝑢 ∈ 𝑣 ↔ 𝑢 ∈ 𝑥))
4	3	imbi1d 343	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑥 → ((𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦) ↔ (𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑦)))
5	4	albidv 1928	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → (∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦) ↔ ∀𝑢(𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑦)))
6	2, 5	imbi12d 346	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → ((𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) ↔ (𝑥 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑦))))
7	6	spvv 1996	. . . . . . . 8 ⊢ (∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) → (𝑥 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑦)))
8		elequ1 2128	. . . . . . . . . 10 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
9		elequ1 2128	. . . . . . . . . 10 ⊢ (𝑢 = 𝑤 → (𝑢 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
10	8, 9	imbi12d 346	. . . . . . . . 9 ⊢ (𝑢 = 𝑤 → ((𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑦) ↔ (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)))
11	10	spvv 1996	. . . . . . . 8 ⊢ (∀𝑢(𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦))
12	7, 11	syl6 35	. . . . . . 7 ⊢ (∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) → (𝑥 ∈ 𝑦 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦)))
13		elequ1 2128	. . . . . . . . . 10 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
14		elequ2 2136	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → (𝑢 ∈ 𝑣 ↔ 𝑢 ∈ 𝑤))
15	14	imbi1d 343	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → ((𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦) ↔ (𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑦)))
16	15	albidv 1928	. . . . . . . . . 10 ⊢ (𝑣 = 𝑤 → (∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦) ↔ ∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑦)))
17	13, 16	imbi12d 346	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) ↔ (𝑤 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑦))))
18	17	spvv 1996	. . . . . . . 8 ⊢ (∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) → (𝑤 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑦)))
19		elequ1 2128	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
20		elequ1 2128	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
21	19, 20	imbi12d 346	. . . . . . . . 9 ⊢ (𝑢 = 𝑧 → ((𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑦) ↔ (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦)))
22	21	spvv 1996	. . . . . . . 8 ⊢ (∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑦) → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦))
23	18, 22	syl6 35	. . . . . . 7 ⊢ (∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) → (𝑤 ∈ 𝑦 → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦)))
24	12, 23	syl6d 75	. . . . . 6 ⊢ (∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦)) → (𝑥 ∈ 𝑦 → (𝑤 ∈ 𝑥 → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦))))
25	24	impcom 409	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦))) → (𝑤 ∈ 𝑥 → (𝑧 ∈ 𝑤 → 𝑧 ∈ 𝑦)))
26	25	impcomd 413	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦))) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
27	26	exlimdv 1941	. . 3 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦))) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
28	27	alrimiv 1935	. 2 ⊢ ((𝑥 ∈ 𝑦 ∧ ∀𝑣(𝑣 ∈ 𝑦 → ∀𝑢(𝑢 ∈ 𝑣 → 𝑢 ∈ 𝑦))) → ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
29	1, 28	eximii 1845	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1546  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-tco 36713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788

This theorem is referenced by: (None)