Theorem axuntco

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

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

Proof

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