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Theorem ismnu 44280
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 44306 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 44318, except for the 𝑥𝑦 clause.

A minimal universe is closed under subsets (mnussd 44282), powersets (mnupwd 44286), and an operation which is similar to a combination of collection and union (mnuop3d 44290), from which closure under pairing (mnuprd 44295), unions (mnuunid 44296), and function ranges (mnurnd 44302) can be deduced, from which equivalence with Grothendieck universes (grumnueq 44306) can be deduced. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypothesis
Ref Expression
ismnu.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
Assertion
Ref Expression
ismnu (𝑈𝑉 → (𝑈𝑀 ↔ ∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
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
StepHypRef Expression
1 simpr 484 . . . . . 6 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝑙 = 𝑧)
21pweqd 4617 . . . . 5 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3 simpl 482 . . . . 5 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝑘 = 𝑈)
42, 3sseq12d 4017 . . . 4 ((𝑘 = 𝑈𝑙 = 𝑧) → (𝒫 𝑙𝑘 ↔ 𝒫 𝑧𝑈))
523adant3 1133 . . . . . . . . . 10 ((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) → 𝒫 𝑙 = 𝒫 𝑧)
65adantr 480 . . . . . . . . 9 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝒫 𝑙 = 𝒫 𝑧)
7 simpr 484 . . . . . . . . 9 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑛 = 𝑤)
86, 7sseq12d 4017 . . . . . . . 8 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (𝒫 𝑙𝑛 ↔ 𝒫 𝑧𝑤))
9 simpl3 1194 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑝 = 𝑖)
10 simpr 484 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑞 = 𝑣)
119, 10eleq12d 2835 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑝𝑞𝑖𝑣))
12 simpl13 1251 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑚 = 𝑓)
1310, 12eleq12d 2835 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑞𝑚𝑣𝑓))
1411, 13anbi12d 632 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → ((𝑝𝑞𝑞𝑚) ↔ (𝑖𝑣𝑣𝑓)))
15 simpl11 1249 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑘 = 𝑈)
1614, 15cbvrexdva2 3349 . . . . . . . . . . 11 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) ↔ ∃𝑣𝑈 (𝑖𝑣𝑣𝑓)))
17 simpl3 1194 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑝 = 𝑖)
18 simpr 484 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
1917, 18eleq12d 2835 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝𝑟𝑖𝑢))
2018unieqd 4920 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
21 simpl2 1193 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑛 = 𝑤)
2220, 21sseq12d 4017 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ( 𝑟𝑛 𝑢𝑤))
2319, 22anbi12d 632 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ((𝑝𝑟 𝑟𝑛) ↔ (𝑖𝑢 𝑢𝑤)))
24 simpl13 1251 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑚 = 𝑓)
2523, 24cbvrexdva2 3349 . . . . . . . . . . 11 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → (∃𝑟𝑚 (𝑝𝑟 𝑟𝑛) ↔ ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))
2616, 25imbi12d 344 . . . . . . . . . 10 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → ((∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
27263expa 1119 . . . . . . . . 9 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → ((∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
28 simpll2 1214 . . . . . . . . 9 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → 𝑙 = 𝑧)
2927, 28cbvraldva2 3348 . . . . . . . 8 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
308, 29anbi12d 632 . . . . . . 7 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → ((𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
31 simpl1 1192 . . . . . . 7 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑘 = 𝑈)
3230, 31cbvrexdva2 3349 . . . . . 6 ((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) → (∃𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ ∃𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
33323expa 1119 . . . . 5 (((𝑘 = 𝑈𝑙 = 𝑧) ∧ 𝑚 = 𝑓) → (∃𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ ∃𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
3433cbvaldvaw 2037 . . . 4 ((𝑘 = 𝑈𝑙 = 𝑧) → (∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
354, 34anbi12d 632 . . 3 ((𝑘 = 𝑈𝑙 = 𝑧) → ((𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)))) ↔ (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
3635, 3cbvraldva2 3348 . 2 (𝑘 = 𝑈 → (∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)))) ↔ ∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
37 ismnu.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
3836, 37elab2g 3680 1 (𝑈𝑉 → (𝑈𝑀 ↔ ∀𝑧𝑈 (𝒫 𝑧𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  wss 3951  𝒫 cpw 4600   cuni 4907
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908
This theorem is referenced by:  mnuop123d  44281  grumnudlem  44304  rr-grothprimbi  44314  rr-groth  44318  dfuniv2  44321
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