Theorem ismnu 44498

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 44524 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 44536, except for the 𝑥 ∈ 𝑦 clause.

A minimal universe is closed under subsets (mnussd 44500), powersets (mnupwd 44504), and an operation which is similar to a combination of collection and union (mnuop3d 44508), from which closure under pairing (mnuprd 44513), unions (mnuunid 44514), and function ranges (mnurnd 44520) can be deduced, from which equivalence with Grothendieck universes (grumnueq 44524) 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 4571	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3		simpl 482	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑘 = 𝑈)
4	2, 3	sseq12d 3967	. . . 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 3967	. . . . . . . 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 2830	. . . . . . . . . . . . 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 2830	. . . . . . . . . . . . 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 3319	. . . . . . . . . . 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 2830	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝 ∈ 𝑟 ↔ 𝑖 ∈ 𝑢))
20	18	unieqd 4876	. . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . 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 3319	. . . . . . . . . . 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 3318	. . . . . . . 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 3319	. . . . . 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 2039	. . . 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 3318	. 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 3635	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 1539   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-v 3442  df-ss 3918  df-pw 4556  df-uni 4864

This theorem is referenced by:  mnuop123d  44499  grumnudlem  44522  rr-grothprimbi  44532  rr-groth  44536  dfuniv2  44539