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Theorem mnugrud 43644
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 43640 . 2 (𝜑 → Tr 𝑈)
42adantr 480 . . . . 5 ((𝜑𝑥𝑈) → 𝑈𝑀)
5 simpr 484 . . . . 5 ((𝜑𝑥𝑈) → 𝑥𝑈)
61, 4, 5mnupwd 43627 . . . 4 ((𝜑𝑥𝑈) → 𝒫 𝑥𝑈)
72ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑈𝑀)
85adantr 480 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑥𝑈)
9 simpr 484 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑦𝑈)
101, 7, 8, 9mnuprd 43636 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → {𝑥, 𝑦} ∈ 𝑈)
1110ralrimiva 3141 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
122ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑈𝑀)
135adantr 480 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑥𝑈)
14 elmapi 8859 . . . . . . . 8 (𝑦 ∈ (𝑈m 𝑥) → 𝑦:𝑥𝑈)
1514adantl 481 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑦:𝑥𝑈)
161, 12, 13, 15mnurnd 43643 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
171, 12, 16mnuunid 43637 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
1817ralrimiva 3141 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)
196, 11, 183jca 1126 . . 3 ((𝜑𝑥𝑈) → (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
2019ralrimiva 3141 . 2 (𝜑 → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
21 elgrug 10807 . . 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 712 1 (𝜑𝑈 ∈ Univ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  {cab 2704  wral 3056  wrex 3065  wss 3944  𝒫 cpw 4598  {cpr 4626   cuni 4903  Tr wtr 5259  ran crn 5673  wf 6538  (class class class)co 7414  m cmap 8836  Univcgru 10805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-fr 5627  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8838  df-gru 10806
This theorem is referenced by:  grumnueq  43647
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