Theorem ismnuprim 45032

Description: Express the predicate on 𝑈 in ismnu 44999 using only primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Assertion

Ref	Expression
ismnuprim	⊢ (∀𝑧 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑧(𝑧 ∈ 𝑈 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))

Distinct variable groups:   𝑣,𝑢   𝑣,𝑖   𝑢,𝑜   𝑈,𝑓   𝑧,𝑤,𝑣   𝑧,𝑡,𝑣   𝑧,𝑓,𝑣   𝑤,𝑈,𝑣   𝑤,𝑜,𝑠

Allowed substitution hints:   𝑈(𝑧, 𝑢, 𝑡, 𝑖, 𝑜, 𝑠)

Proof of Theorem ismnuprim

Step	Hyp	Ref	Expression
1		19.28v 2025	. . 3 ⊢ (∀𝑓(𝒫 𝑧 ⊆ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
2		r19.42v 3196	. . . . 5 ⊢ (∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
3		19.26 1899	. . . . . . . 8 ⊢ (∀𝑣((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑣∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
4		19.26 1899	. . . . . . . . . . 11 ⊢ (∀𝑣((𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ (𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ↔ (∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
5		jcab 526	. . . . . . . . . . . 12 ⊢ ((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ ((𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ (𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
6	5	albii 1848	. . . . . . . . . . 11 ⊢ (∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ ∀𝑣((𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ (𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
7		pwss 4585	. . . . . . . . . . . 12 ⊢ (𝒫 𝑧 ⊆ 𝑈 ↔ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈))
8		pwss 4585	. . . . . . . . . . . 12 ⊢ (𝒫 𝑧 ⊆ 𝑤 ↔ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤))
9	7, 8	anbi12i 639	. . . . . . . . . . 11 ⊢ ((𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤) ↔ (∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
10	4, 6, 9	3bitr4i 306	. . . . . . . . . 10 ⊢ (∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤))
11		ralcom4 3290	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝑧 ∀𝑣((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ ∀𝑣∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
12		19.23v 1971	. . . . . . . . . . . . 13 ⊢ (∀𝑣((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
13		3anass 1110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ (𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
14	13	exbii 1877	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
15		df-rex 3089	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
16	14, 15	bitr4i 281	. . . . . . . . . . . . . 14 ⊢ (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ ∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓))
17	16	imbi1i 352	. . . . . . . . . . . . 13 ⊢ ((∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
18	12, 17	bitri 278	. . . . . . . . . . . 12 ⊢ (∀𝑣((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
19	18	ralbii 3110	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝑧 ∀𝑣((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
20	11, 19	bitr3i 280	. . . . . . . . . 10 ⊢ (∀𝑣∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
21	10, 20	anbi12i 639	. . . . . . . . 9 ⊢ ((∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑣∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ ((𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
22		anass 473	. . . . . . . . 9 ⊢ (((𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
23	21, 22	bitri 278	. . . . . . . 8 ⊢ ((∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑣∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
24	3, 23	bitri 278	. . . . . . 7 ⊢ (∀𝑣((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
25		df-ss 3921	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑧 ↔ ∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧))
26		df-an 401	. . . . . . . . . 10 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤) ↔ ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤))
27	25, 26	imbi12i 353	. . . . . . . . 9 ⊢ ((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ (∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)))
28		3impexp 1376	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
29		biid 264	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑢 ↔ 𝑖 ∈ 𝑢)
30		expanduniss 45031	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑢 ⊆ 𝑤 ↔ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))
31	29, 30	expandan 45026	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ¬ (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))
32	31	expandrexn 45029	. . . . . . . . . . . . . 14 ⊢ (∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))
33	32	imbi2i 339	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ 𝑓 → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))
34	33	imbi2i 339	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))
35	34	imbi2i 339	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))
36	28, 35	bitri 278	. . . . . . . . . 10 ⊢ (((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))
37	36	expandral 45028	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))
38	27, 37	expandan 45026	. . . . . . . 8 ⊢ (((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))
39	38	albii 1848	. . . . . . 7 ⊢ (∀𝑣((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ∧ ∀𝑖 ∈ 𝑧 ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))
40	24, 39	bitr3i 280	. . . . . 6 ⊢ ((𝒫 𝑧 ⊆ 𝑈 ∧ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))
41	40	expandrex 45030	. . . . 5 ⊢ (∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))
42	2, 41	bitr3i 280	. . . 4 ⊢ ((𝒫 𝑧 ⊆ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))
43	42	albii 1848	. . 3 ⊢ (∀𝑓(𝒫 𝑧 ⊆ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))
44	1, 43	bitr3i 280	. 2 ⊢ ((𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))))))))
45	44	expandral 45028	1 ⊢ (∀𝑧 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ↔ ∀𝑧(𝑧 ∈ 𝑈 → ∀𝑓 ¬ ∀𝑤(𝑤 ∈ 𝑈 → ¬ ∀𝑣 ¬ ((∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧) → ¬ (𝑣 ∈ 𝑈 → ¬ 𝑣 ∈ 𝑤)) → ¬ ∀𝑖(𝑖 ∈ 𝑧 → (𝑣 ∈ 𝑈 → (𝑖 ∈ 𝑣 → (𝑣 ∈ 𝑓 → ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))))))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567  ∃wex 1808   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871

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-9 2152  ax-11 2191  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3456  df-ss 3921  df-pw 4563  df-uni 4872

This theorem is used by:  rr-grothprimbi  45033