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Theorem mnugrud 44303
Description: Minimal universes are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnugrud.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnugrud.2 (𝜑𝑈𝑀)
Assertion
Ref Expression
mnugrud (𝜑𝑈 ∈ Univ)
Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,𝑘,𝑚,𝑛,𝑝,𝑙
Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnugrud
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnugrud.1 . . 3 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnugrud.2 . . 3 (𝜑𝑈𝑀)
31, 2mnutrd 44299 . 2 (𝜑 → Tr 𝑈)
42adantr 480 . . . . 5 ((𝜑𝑥𝑈) → 𝑈𝑀)
5 simpr 484 . . . . 5 ((𝜑𝑥𝑈) → 𝑥𝑈)
61, 4, 5mnupwd 44286 . . . 4 ((𝜑𝑥𝑈) → 𝒫 𝑥𝑈)
72ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑈𝑀)
85adantr 480 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑥𝑈)
9 simpr 484 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑦𝑈)
101, 7, 8, 9mnuprd 44295 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → {𝑥, 𝑦} ∈ 𝑈)
1110ralrimiva 3146 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
122ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑈𝑀)
135adantr 480 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑥𝑈)
14 elmapi 8889 . . . . . . . 8 (𝑦 ∈ (𝑈m 𝑥) → 𝑦:𝑥𝑈)
1514adantl 481 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑦:𝑥𝑈)
161, 12, 13, 15mnurnd 44302 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
171, 12, 16mnuunid 44296 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
1817ralrimiva 3146 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)
196, 11, 183jca 1129 . . 3 ((𝜑𝑥𝑈) → (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
2019ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
21 elgrug 10832 . . 3 (𝑈𝑀 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
222, 21syl 17 . 2 (𝜑 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))))
233, 20, 22mpbir2and 713 1 (𝜑𝑈 ∈ Univ)
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  {cpr 4628   cuni 4907  Tr wtr 5259  ran crn 5686  wf 6557  (class class class)co 7431  m cmap 8866  Univcgru 10830
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-fr 5637  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-gru 10831
This theorem is referenced by:  grumnueq  44306
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