Theorem ismnu 44257

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 44283 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 44295, except for the 𝑥 ∈ 𝑦 clause.

A minimal universe is closed under subsets (mnussd 44259), powersets (mnupwd 44263), and an operation which is similar to a combination of collection and union (mnuop3d 44267), from which closure under pairing (mnuprd 44272), unions (mnuunid 44273), and function ranges (mnurnd 44279) can be deduced, from which equivalence with Grothendieck universes (grumnueq 44283) 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 484	. . . . . 6 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑙 = 𝑧)
2	1	pweqd 4583	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3		simpl 482	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑘 = 𝑈)
4	2, 3	sseq12d 3983	. . . 4 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → (𝒫 𝑙 ⊆ 𝑘 ↔ 𝒫 𝑧 ⊆ 𝑈))
5	2	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) → 𝒫 𝑙 = 𝒫 𝑧)
6	5	adantr 480	. . . . . . . . 9 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝒫 𝑙 = 𝒫 𝑧)
7		simpr 484	. . . . . . . . 9 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑛 = 𝑤)
8	6, 7	sseq12d 3983	. . . . . . . 8 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (𝒫 𝑙 ⊆ 𝑛 ↔ 𝒫 𝑧 ⊆ 𝑤))
9		simpl3 1194	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑝 = 𝑖)
10		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑞 = 𝑣)
11	9, 10	eleq12d 2823	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑝 ∈ 𝑞 ↔ 𝑖 ∈ 𝑣))
12		simpl13 1251	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑚 = 𝑓)
13	10, 12	eleq12d 2823	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑞 ∈ 𝑚 ↔ 𝑣 ∈ 𝑓))
14	11, 13	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → ((𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) ↔ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓)))
15		simpl11 1249	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑘 = 𝑈)
16	14, 15	cbvrexdva2 3324	. . . . . . . . . . 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 1194	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑝 = 𝑖)
18		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
19	17, 18	eleq12d 2823	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝 ∈ 𝑟 ↔ 𝑖 ∈ 𝑢))
20	18	unieqd 4887	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ∪ 𝑟 = ∪ 𝑢)
21		simpl2 1193	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑛 = 𝑤)
22	20, 21	sseq12d 3983	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (∪ 𝑟 ⊆ 𝑛 ↔ ∪ 𝑢 ⊆ 𝑤))
23	19, 22	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ((𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛) ↔ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
24		simpl13 1251	. . . . . . . . . . . 12 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑚 = 𝑓)
25	23, 24	cbvrexdva2 3324	. . . . . . . . . . 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 344	. . . . . . . . . 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 1118	. . . . . . . . 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 1214	. . . . . . . . 9 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → 𝑙 = 𝑧)
29	27, 28	cbvraldva2 3323	. . . . . . . 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 632	. . . . . . 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 1192	. . . . . . 7 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑘 = 𝑈)
32	30, 31	cbvrexdva2 3324	. . . . . 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 1118	. . . . 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 2038	. . . 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 632	. . 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 3323	. 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 3650	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))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-v 3452  df-ss 3934  df-pw 4568  df-uni 4875

This theorem is referenced by:  mnuop123d  44258  grumnudlem  44281  rr-grothprimbi  44291  rr-groth  44295  dfuniv2  44298