Theorem ismnu 45085

Description: The hypothesis of this theorem defines a class M of sets that we temporarily call "minimal universes", and which will turn out in grumnueq 45111 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 45123, except for the 𝑥 ∈ 𝑦 clause.

A minimal universe is closed under subsets (mnussd 45087), powersets (mnupwd 45091), and an operation which is similar to a combination of collection and union (mnuop3d 45095), from which closure under pairing (mnuprd 45100), unions (mnuunid 45101), and function ranges (mnurnd 45107) can be deduced, from which equivalence with Grothendieck universes (grumnueq 45111) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypothesis

Ref	Expression
ismnu.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}

Assertion

Ref	Expression
ismnu	⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ 𝑀 ↔ ∀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))))))

Distinct variable groups:   𝑧,𝑤,𝑣,𝑈,𝑓,𝑖,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙   𝑧,𝑢,𝑟,𝑤,𝑈,𝑓,𝑖,𝑘,𝑚,𝑛,𝑝,𝑙

Allowed substitution hints:   𝑀(𝑧, 𝑤, 𝑣, 𝑢, 𝑓, 𝑖, 𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)   𝑉(𝑧, 𝑤, 𝑣, 𝑢, 𝑓, 𝑖, 𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)

Proof of Theorem ismnu

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑙 = 𝑧)
2	1	pweqd 4574	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3		simpl 488	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑘 = 𝑈)
4	2, 3	sseq12d 3964	. . . 4 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → (𝒫 𝑙 ⊆ 𝑘 ↔ 𝒫 𝑧 ⊆ 𝑈))
5	2	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) → 𝒫 𝑙 = 𝒫 𝑧)
6	5	adantr 486	. . . . . . . . 9 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝒫 𝑙 = 𝒫 𝑧)
7		simpr 490	. . . . . . . . 9 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑛 = 𝑤)
8	6, 7	sseq12d 3964	. . . . . . . 8 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (𝒫 𝑙 ⊆ 𝑛 ↔ 𝒫 𝑧 ⊆ 𝑤))
9		simpl3 1212	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑝 = 𝑖)
10		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑞 = 𝑣)
11	9, 10	eleq12d 2854	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑝 ∈ 𝑞 ↔ 𝑖 ∈ 𝑣))
12		simpl13 1269	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑚 = 𝑓)
13	10, 12	eleq12d 2854	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑞 ∈ 𝑚 ↔ 𝑣 ∈ 𝑓))
14	11, 13	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → ((𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) ↔ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
15		simpl11 1267	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑘 = 𝑈)
16	14, 15	cbvrexdva2 3337	. . . . . . . . . . 11 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) → (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) ↔ ∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
17		simpl3 1212	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑝 = 𝑖)
18		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
19	17, 18	eleq12d 2854	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝 ∈ 𝑟 ↔ 𝑖 ∈ 𝑢))
20	18	unieqd 4880	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ∪ 𝑟 = ∪ 𝑢)
21		simpl2 1211	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑛 = 𝑤)
22	20, 21	sseq12d 3964	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (∪ 𝑟 ⊆ 𝑛 ↔ ∪ 𝑢 ⊆ 𝑤))
23	19, 22	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ((𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛) ↔ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
24		simpl13 1269	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑚 = 𝑓)
25	23, 24	cbvrexdva2 3337	. . . . . . . . . . 11 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) → (∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛) ↔ ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
26	16, 25	imbi12d 347	. . . . . . . . . 10 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) → ((∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛)) ↔ (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
27	26	3expa 1136	. . . . . . . . 9 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → ((∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛)) ↔ (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
28		simpll2 1232	. . . . . . . . 9 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → 𝑙 = 𝑧)
29	27, 28	cbvraldva2 3336	. . . . . . . 8 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛)) ↔ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
30	8, 29	anbi12d 644	. . . . . . 7 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → ((𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))) ↔ (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
31		simpl1 1210	. . . . . . 7 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑘 = 𝑈)
32	30, 31	cbvrexdva2 3337	. . . . . 6 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) → (∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))) ↔ ∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
33	32	3expa 1136	. . . . 5 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) ∧ 𝑚 = 𝑓) → (∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))) ↔ ∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
34	33	cbvaldvaw 2071	. . . 4 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → (∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))) ↔ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
35	4, 34	anbi12d 644	. . 3 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → ((𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛)))) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))))
36	35, 3	cbvraldva2 3336	. 2 ⊢ (𝑘 = 𝑈 → (∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛)))) ↔ ∀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))))))
37		ismnu.1	. 2 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
38	36, 37	elab2g 3634	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ 𝑀 ↔ ∀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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3916  df-pw 4559  df-uni 4868

This theorem is used by:  mnuop123d  45086  grumnudlem  45109  rr-grothprimbi  45119  rr-groth  45123  dfuniv2  45126