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Theorem mnugrud 41902
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 41898 . 2 (𝜑 → Tr 𝑈)
42adantr 481 . . . . 5 ((𝜑𝑥𝑈) → 𝑈𝑀)
5 simpr 485 . . . . 5 ((𝜑𝑥𝑈) → 𝑥𝑈)
61, 4, 5mnupwd 41885 . . . 4 ((𝜑𝑥𝑈) → 𝒫 𝑥𝑈)
72ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑈𝑀)
85adantr 481 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑥𝑈)
9 simpr 485 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑦𝑈)
101, 7, 8, 9mnuprd 41894 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → {𝑥, 𝑦} ∈ 𝑈)
1110ralrimiva 3103 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
122ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑈𝑀)
135adantr 481 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑥𝑈)
14 elmapi 8637 . . . . . . . 8 (𝑦 ∈ (𝑈m 𝑥) → 𝑦:𝑥𝑈)
1514adantl 482 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑦:𝑥𝑈)
161, 12, 13, 15mnurnd 41901 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
171, 12, 16mnuunid 41895 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
1817ralrimiva 3103 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)
196, 11, 183jca 1127 . . 3 ((𝜑𝑥𝑈) → (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
2019ralrimiva 3103 . 2 (𝜑 → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
21 elgrug 10548 . . 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 710 1 (𝜑𝑈 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wal 1537   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wrex 3065  wss 3887  𝒫 cpw 4533  {cpr 4563   cuni 4839  Tr wtr 5191  ran crn 5590  wf 6429  (class class class)co 7275  m cmap 8615  Univcgru 10546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-fr 5544  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-gru 10547
This theorem is referenced by:  grumnueq  41905
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