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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 𝑥𝑦 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 ∣ ∀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 4622 . . . . 5 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3 simpl 482 . . . . 5 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝑘 = 𝑈)
42, 3sseq12d 4029 . . . 4 ((𝑘 = 𝑈𝑙 = 𝑧) → (𝒫 𝑙𝑘 ↔ 𝒫 𝑧𝑈))
523adant3 1131 . . . . . . . . . 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 4029 . . . . . . . 8 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (𝒫 𝑙𝑛 ↔ 𝒫 𝑧𝑤))
9 simpl3 1192 . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑝𝑞𝑖𝑣))
12 simpl13 1249 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑚 = 𝑓)
1310, 12eleq12d 2833 . . . . . . . . . . . . 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 1247 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑘 = 𝑈)
1614, 15cbvrexdva2 3347 . . . . . . . . . . 11 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) ↔ ∃𝑣𝑈 (𝑖𝑣𝑣𝑓)))
17 simpl3 1192 . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝𝑟𝑖𝑢))
2018unieqd 4925 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
21 simpl2 1191 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑛 = 𝑤)
2220, 21sseq12d 4029 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑚 = 𝑓)
2523, 24cbvrexdva2 3347 . . . . . . . . . . 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 1117 . . . . . . . . 9 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → ((∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
28 simpll2 1212 . . . . . . . . 9 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → 𝑙 = 𝑧)
2927, 28cbvraldva2 3346 . . . . . . . 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 1190 . . . . . . 7 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑘 = 𝑈)
3230, 31cbvrexdva2 3347 . . . . . 6 ((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) → (∃𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ ∃𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
33323expa 1117 . . . . 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 2035 . . . 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 3346 . 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 3683 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 1086  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  𝒫 cpw 4605   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-ss 3980  df-pw 4607  df-uni 4913
This theorem is referenced by:  mnuop123d  44258  grumnudlem  44281  rr-grothprimbi  44291  rr-groth  44295  dfuniv2  44298
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