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Theorem mnugrud 43498
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 43494 . 2 (𝜑 → Tr 𝑈)
42adantr 480 . . . . 5 ((𝜑𝑥𝑈) → 𝑈𝑀)
5 simpr 484 . . . . 5 ((𝜑𝑥𝑈) → 𝑥𝑈)
61, 4, 5mnupwd 43481 . . . 4 ((𝜑𝑥𝑈) → 𝒫 𝑥𝑈)
72ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑈𝑀)
85adantr 480 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑥𝑈)
9 simpr 484 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑦𝑈)
101, 7, 8, 9mnuprd 43490 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → {𝑥, 𝑦} ∈ 𝑈)
1110ralrimiva 3138 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
122ad2antrr 723 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑈𝑀)
135adantr 480 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑥𝑈)
14 elmapi 8838 . . . . . . . 8 (𝑦 ∈ (𝑈m 𝑥) → 𝑦:𝑥𝑈)
1514adantl 481 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑦:𝑥𝑈)
161, 12, 13, 15mnurnd 43497 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
171, 12, 16mnuunid 43491 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
1817ralrimiva 3138 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈)
196, 11, 183jca 1125 . . 3 ((𝜑𝑥𝑈) → (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
2019ralrimiva 3138 . 2 (𝜑 → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
21 elgrug 10782 . . 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 395  w3a 1084  wal 1531   = wceq 1533  wcel 2098  {cab 2701  wral 3053  wrex 3062  wss 3940  𝒫 cpw 4594  {cpr 4622   cuni 4899  Tr wtr 5255  ran crn 5667  wf 6529  (class class class)co 7401  m cmap 8815  Univcgru 10780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-reg 9582
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-fr 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8817  df-gru 10781
This theorem is referenced by:  grumnueq  43501
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