Theorem ismnuprim 40679

Description: Express the predicate on 𝑈 in ismnu 40646 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 1997	. . 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 3350	. . . . 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 1871	. . . . . . . 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 1871	. . . . . . . . . . 11 ⊢ (∀𝑣((𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ (𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ↔ (∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
5		jcab 520	. . . . . . . . . . . 12 ⊢ ((𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ ((𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ (𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
6	5	albii 1820	. . . . . . . . . . 11 ⊢ (∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ ∀𝑣((𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ (𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
7		pwss 4564	. . . . . . . . . . . 12 ⊢ (𝒫 𝑧 ⊆ 𝑈 ↔ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈))
8		pwss 4564	. . . . . . . . . . . 12 ⊢ (𝒫 𝑧 ⊆ 𝑤 ↔ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤))
9	7, 8	anbi12i 628	. . . . . . . . . . 11 ⊢ ((𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤) ↔ (∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑈) ∧ ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)))
10	4, 6, 9	3bitr4i 305	. . . . . . . . . 10 ⊢ (∀𝑣(𝑣 ⊆ 𝑧 → (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝑤)) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ 𝒫 𝑧 ⊆ 𝑤))
11		ralcom4 3235	. . . . . . . . . . 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 1943	. . . . . . . . . . . . 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 1091	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ (𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
14	13	exbii 1848	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
15		df-rex 3144	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
16	14, 15	bitr4i 280	. . . . . . . . . . . . . 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 277	. . . . . . . . . . . 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 3165	. . . . . . . . . . 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 279	. . . . . . . . . 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 628	. . . . . . . . 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 471	. . . . . . . . 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 277	. . . . . . . 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 277	. . . . . . 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		dfss2 3955	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑧 ↔ ∀𝑡(𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧))
26		df-an 399	. . . . . . . . . 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 1354	. . . . . . . . . . 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 263	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑢 ↔ 𝑖 ∈ 𝑢)
30		expanduniss 40678	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑢 ⊆ 𝑤 ↔ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))
31	29, 30	expandan 40673	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ¬ (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤))))
32	31	expandrexn 40676	. . . . . . . . . . . . . 14 ⊢ (∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ ¬ ∀𝑢(𝑢 ∈ 𝑓 → (𝑖 ∈ 𝑢 → ¬ ∀𝑜(𝑜 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤)))))
33	32	imbi2i 338	. . . . . . . . . . . . 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 338	. . . . . . . . . . . 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 338	. . . . . . . . . . 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 277	. . . . . . . . . 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 40675	. . . . . . . . 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 40673	. . . . . . . 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 1820	. . . . . . 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 279	. . . . . 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 40677	. . . . 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 279	. . . 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 1820	. . 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 279	. 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 40675	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))))))))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3943  df-ss 3952  df-pw 4541  df-uni 4839

This theorem is referenced by:  rr-grothprimbi  40680