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Theorem mnussd 40891
 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 40890 . . 3 (𝜑 → (𝒫 𝐴𝑈 ∧ ∀𝑓𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝑓) → ∃𝑢𝑓 (𝑖𝑢 𝑢𝑤)))))
54simpld 498 . 2 (𝜑 → 𝒫 𝐴𝑈)
6 mnussd.4 . . 3 (𝜑𝐵𝐴)
73, 6sselpwd 5216 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
85, 7sseldd 3954 1 (𝜑𝐵𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2115  {cab 2802  ∀wral 3133  ∃wrex 3134   ⊆ wss 3919  𝒫 cpw 4522  ∪ cuni 4824 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 2179  ax-ext 2796  ax-sep 5189 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 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524  df-uni 4825 This theorem is referenced by:  mnuss2d  40892  mnu0eld  40893  mnusnd  40896  mnuprssd  40897  mnuprdlem4  40903  mnutrcld  40907
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