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Theorem mnussd 43012
Description: Minimal universes are closed under subsets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnussd.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnussd.2 (𝜑𝑈𝑀)
mnussd.3 (𝜑𝐴𝑈)
mnussd.4 (𝜑𝐵𝐴)
Assertion
Ref Expression
mnussd (𝜑𝐵𝑈)
Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙   𝑈,𝑟,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 mnussd
Dummy variables 𝑤 𝑓 𝑖 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnussd.1 . . . 4 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnussd.2 . . . 4 (𝜑𝑈𝑀)
3 mnussd.3 . . . 4 (𝜑𝐴𝑈)
41, 2, 3mnuop123d 43011 . . 3 (𝜑 → (𝒫 𝐴𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
54simpld 495 . 2 (𝜑 → 𝒫 𝐴𝑈)
6 mnussd.4 . . 3 (𝜑𝐵𝐴)
73, 6sselpwd 5326 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
85, 7sseldd 3983 1 (𝜑𝐵𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wcel 2106  {cab 2709  wral 3061  wrex 3070  wss 3948  𝒫 cpw 4602   cuni 4908
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909
This theorem is referenced by:  mnuss2d  43013  mnu0eld  43014  mnusnd  43017  mnuprssd  43018  mnuprdlem4  43024  mnutrcld  43028
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