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Theorem mnutrd 40773
 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 484 . . . . 5 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑈𝑀)
4 simprr 772 . . . . 5 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑦𝑈)
5 simprl 770 . . . . 5 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑥𝑦)
61, 3, 4, 5mnutrcld 40772 . . . 4 ((𝜑 ∧ (𝑥𝑦𝑦𝑈)) → 𝑥𝑈)
76ex 416 . . 3 (𝜑 → ((𝑥𝑦𝑦𝑈) → 𝑥𝑈))
87alrimivv 1930 . 2 (𝜑 → ∀𝑥𝑦((𝑥𝑦𝑦𝑈) → 𝑥𝑈))
9 dftr2 5147 . 2 (Tr 𝑈 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑈) → 𝑥𝑈))
108, 9sylibr 237 1 (𝜑 → Tr 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2115  {cab 2799  ∀wral 3126  ∃wrex 3127   ⊆ wss 3910  𝒫 cpw 4512  ∪ cuni 4811  Tr wtr 5145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-in 3917  df-ss 3927  df-pw 4514  df-sn 4541  df-uni 4812  df-tr 5146 This theorem is referenced by:  mnugrud  40777
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