Theorem ismnu 41421

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 41447 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 41459, except for the 𝑥 ∈ 𝑦 clause.

A minimal universe is closed under subsets (mnussd 41423), powersets (mnupwd 41427), and an operation which is similar to a combination of collection and union (mnuop3d 41431), from which closure under pairing (mnuprd 41436), unions (mnuunid 41437), and function ranges (mnurnd 41443) can be deduced, from which equivalence with Grothendieck universes (grumnueq 41447) 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 488	. . . . . 6 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑙 = 𝑧)
2	1	pweqd 4507	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3		simpl 486	. . . . 5 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → 𝑘 = 𝑈)
4	2, 3	sseq12d 3910	. . . 4 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧) → (𝒫 𝑙 ⊆ 𝑘 ↔ 𝒫 𝑧 ⊆ 𝑈))
5	2	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) → 𝒫 𝑙 = 𝒫 𝑧)
6	5	adantr 484	. . . . . . . . 9 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝒫 𝑙 = 𝒫 𝑧)
7		simpr 488	. . . . . . . . 9 ⊢ (((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑛 = 𝑤)
8	6, 7	sseq12d 3910	. . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑞 = 𝑣)
11	9, 10	eleq12d 2827	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑞 ∈ 𝑚 ↔ 𝑣 ∈ 𝑓))
14	11, 13	anbi12d 634	. . . . . . . . . . . 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 3359	. . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
19	17, 18	eleq12d 2827	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝 ∈ 𝑟 ↔ 𝑖 ∈ 𝑢))
20	18	unieqd 4810	. . . . . . . . . . . . . 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 3910	. . . . . . . . . . . . 13 ⊢ ((((𝑘 = 𝑈 ∧ 𝑙 = 𝑧 ∧ 𝑚 = 𝑓) ∧ 𝑛 = 𝑤 ∧ 𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (∪ 𝑟 ⊆ 𝑛 ↔ ∪ 𝑢 ⊆ 𝑤))
23	19, 22	anbi12d 634	. . . . . . . . . . . 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 3359	. . . . . . . . . . 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 348	. . . . . . . . . 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 1119	. . . . . . . . 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 3358	. . . . . . . 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 634	. . . . . . 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 3359	. . . . . 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 1119	. . . . 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 2050	. . . 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 634	. . 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 3358	. 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 3575	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 209   ∧ wa 399   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2716  ∀wral 3053  ∃wrex 3054   ⊆ wss 3843  𝒫 cpw 4488  ∪ cuni 4796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-in 3850  df-ss 3860  df-pw 4490  df-uni 4797

This theorem is referenced by:  mnuop123d  41422  grumnudlem  41445  rr-grothprimbi  41455  rr-groth  41459