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Theorem mnugrud 41485
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 41481 . 2 (𝜑 → Tr 𝑈)
42adantr 484 . . . . 5 ((𝜑𝑥𝑈) → 𝑈𝑀)
5 simpr 488 . . . . 5 ((𝜑𝑥𝑈) → 𝑥𝑈)
61, 4, 5mnupwd 41468 . . . 4 ((𝜑𝑥𝑈) → 𝒫 𝑥𝑈)
72ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑈𝑀)
85adantr 484 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑥𝑈)
9 simpr 488 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → 𝑦𝑈)
101, 7, 8, 9mnuprd 41477 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦𝑈) → {𝑥, 𝑦} ∈ 𝑈)
1110ralrimiva 3097 . . . 4 ((𝜑𝑥𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
122ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑈𝑀)
135adantr 484 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑥𝑈)
14 elmapi 8472 . . . . . . . 8 (𝑦 ∈ (𝑈m 𝑥) → 𝑦:𝑥𝑈)
1514adantl 485 . . . . . . 7 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → 𝑦:𝑥𝑈)
161, 12, 13, 15mnurnd 41484 . . . . . 6 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
171, 12, 16mnuunid 41478 . . . . 5 (((𝜑𝑥𝑈) ∧ 𝑦 ∈ (𝑈m 𝑥)) → ran 𝑦𝑈)
1817ralrimiva 3097 . . . 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 3097 . 2 (𝜑 → ∀𝑥𝑈 (𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈m 𝑥) ran 𝑦𝑈))
21 elgrug 10305 . . 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 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wcel 2114  {cab 2717  wral 3054  wrex 3055  wss 3853  𝒫 cpw 4498  {cpr 4528   cuni 4806  Tr wtr 5146  ran crn 5536  wf 6346  (class class class)co 7183  m cmap 8450  Univcgru 10303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492  ax-reg 9142
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-fr 5493  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-1st 7727  df-2nd 7728  df-map 8452  df-gru 10304
This theorem is referenced by:  grumnueq  41488
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