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Theorem ismnu 40820
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 40846 to be exactly Grothendicek universes. Minimal universes are sets which satisfy the predicate on 𝑦 in rr-groth 40858, except for the 𝑥𝑦 clause.

A minimal universe is closed under subsets (mnussd 40822), powersets (mnupwd 40826), and an operation which is similar to a combination of collection and union (mnuop3d 40830), from which closure under pairing (mnuprd 40835), unions (mnuunid 40836), and function ranges (mnurnd 40842) can be deduced, from which equivalence with Grothendieck universes (grumnueq 40846) 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 488 . . . . . 6 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝑙 = 𝑧)
21pweqd 4539 . . . . 5 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝒫 𝑙 = 𝒫 𝑧)
3 simpl 486 . . . . 5 ((𝑘 = 𝑈𝑙 = 𝑧) → 𝑘 = 𝑈)
42, 3sseq12d 3984 . . . 4 ((𝑘 = 𝑈𝑙 = 𝑧) → (𝒫 𝑙𝑘 ↔ 𝒫 𝑧𝑈))
523adant3 1129 . . . . . . . . . 10 ((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) → 𝒫 𝑙 = 𝒫 𝑧)
65adantr 484 . . . . . . . . 9 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝒫 𝑙 = 𝒫 𝑧)
7 simpr 488 . . . . . . . . 9 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑛 = 𝑤)
86, 7sseq12d 3984 . . . . . . . 8 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (𝒫 𝑙𝑛 ↔ 𝒫 𝑧𝑤))
9 simpl3 1190 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑝 = 𝑖)
10 simpr 488 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑞 = 𝑣)
119, 10eleq12d 2910 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑝𝑞𝑖𝑣))
12 simpl13 1247 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑚 = 𝑓)
1310, 12eleq12d 2910 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → (𝑞𝑚𝑣𝑓))
1411, 13anbi12d 633 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → ((𝑝𝑞𝑞𝑚) ↔ (𝑖𝑣𝑣𝑓)))
15 simpl11 1245 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑞 = 𝑣) → 𝑘 = 𝑈)
1614, 15cbvrexdva2 3442 . . . . . . . . . . 11 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) ↔ ∃𝑣𝑈 (𝑖𝑣𝑣𝑓)))
17 simpl3 1190 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑝 = 𝑖)
18 simpr 488 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
1917, 18eleq12d 2910 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → (𝑝𝑟𝑖𝑢))
2018unieqd 4835 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑟 = 𝑢)
21 simpl2 1189 . . . . . . . . . . . . . 14 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑛 = 𝑤)
2220, 21sseq12d 3984 . . . . . . . . . . . . 13 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ( 𝑟𝑛 𝑢𝑤))
2319, 22anbi12d 633 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → ((𝑝𝑟 𝑟𝑛) ↔ (𝑖𝑢 𝑢𝑤)))
24 simpl13 1247 . . . . . . . . . . . 12 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) ∧ 𝑟 = 𝑢) → 𝑚 = 𝑓)
2523, 24cbvrexdva2 3442 . . . . . . . . . . 11 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → (∃𝑟𝑚 (𝑝𝑟 𝑟𝑛) ↔ ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))
2616, 25imbi12d 348 . . . . . . . . . 10 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤𝑝 = 𝑖) → ((∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
27263expa 1115 . . . . . . . . 9 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → ((∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
28 simpll2 1210 . . . . . . . . 9 ((((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) ∧ 𝑝 = 𝑖) → 𝑙 = 𝑧)
2927, 28cbvraldva2 3441 . . . . . . . 8 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → (∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛)) ↔ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤))))
308, 29anbi12d 633 . . . . . . 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 1188 . . . . . . 7 (((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) ∧ 𝑛 = 𝑤) → 𝑘 = 𝑈)
3230, 31cbvrexdva2 3442 . . . . . 6 ((𝑘 = 𝑈𝑙 = 𝑧𝑚 = 𝑓) → (∃𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ ∃𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
33323expa 1115 . . . . 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 2046 . . . 4 ((𝑘 = 𝑈𝑙 = 𝑧) → (∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))) ↔ ∀𝑓𝑤𝑈 (𝒫 𝑧𝑤 ∧ ∀𝑖𝑧 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
354, 34anbi12d 633 . . 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 3441 . 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 3653 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 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2115  {cab 2802  wral 3132  wrex 3133  wss 3918  𝒫 cpw 4520   cuni 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-v 3481  df-in 3925  df-ss 3935  df-pw 4522  df-uni 4822
This theorem is referenced by:  mnuop123d  40821  grumnudlem  40844  rr-grothprimbi  40854  rr-groth  40858
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