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Theorem mnutrcld 43765
Description: Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnutrcld.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnutrcld.2 (𝜑𝑈𝑀)
mnutrcld.3 (𝜑𝐴𝑈)
mnutrcld.4 (𝜑𝐵𝐴)
Assertion
Ref Expression
mnutrcld (𝜑𝐵𝑈)
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)   𝐴(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐵(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnutrcld
StepHypRef Expression
1 mnutrcld.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnutrcld.2 . 2 (𝜑𝑈𝑀)
3 mnutrcld.3 . . 3 (𝜑𝐴𝑈)
41, 2, 3mnuunid 43763 . 2 (𝜑 𝐴𝑈)
5 mnutrcld.4 . . 3 (𝜑𝐵𝐴)
6 elssuni 4944 . . 3 (𝐵𝐴𝐵 𝐴)
75, 6syl 17 . 2 (𝜑𝐵 𝐴)
81, 2, 4, 7mnussd 43749 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531   = wceq 1533  wcel 2098  {cab 2705  wral 3058  wrex 3067  wss 3949  𝒫 cpw 4606   cuni 4912
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-ext 2699  ax-sep 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966  df-pw 4608  df-sn 4633  df-uni 4913
This theorem is referenced by:  mnutrd  43766  mnurndlem2  43768
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