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Theorem mnutrd 42652
Description: Minimal universes are transitive. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnutrd.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnutrd.2 (𝜑𝑈𝑀)
Assertion
Ref Expression
mnutrd (𝜑 → Tr 𝑈)
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 mnutrd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnutrd.1 . . . . 5 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnutrd.2 . . . . . 6 (𝜑𝑈𝑀)
32adantr 482 . . . . 5 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑈𝑀)
4 simprr 772 . . . . 5 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑦𝑈)
5 simprl 770 . . . . 5 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑥𝑦)
61, 3, 4, 5mnutrcld 42651 . . . 4 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑥𝑈)
76ex 414 . . 3 (𝜑 → ((𝑥𝑦𝑦𝑈) → 𝑥𝑈))
87alrimivv 1932 . 2 (𝜑 → ∀𝑥𝑦((𝑥𝑦𝑦𝑈) → 𝑥𝑈))
9 dftr2 5228 . 2 (Tr 𝑈 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑈) → 𝑥𝑈))
108, 9sylibr 233 1 (𝜑 → Tr 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  wss 3914  𝒫 cpw 4564   cuni 4869  Tr wtr 5226
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566  df-sn 4591  df-uni 4870  df-tr 5227
This theorem is referenced by:  mnugrud  42656
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