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Theorem mnupwd 41885
Description: Minimal universes are closed under powersets. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnupwd.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnupwd.2 (𝜑𝑈𝑀)
mnupwd.3 (𝜑𝐴𝑈)
Assertion
Ref Expression
mnupwd (𝜑 → 𝒫 𝐴𝑈)
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)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnupwd
Dummy variables 𝑤 𝑖 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnupwd.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnupwd.2 . 2 (𝜑𝑈𝑀)
3 mnupwd.3 . . . 4 (𝜑𝐴𝑈)
4 0ex 5231 . . . . 5 ∅ ∈ V
54a1i 11 . . . 4 (𝜑 → ∅ ∈ V)
61, 2, 3, 5mnuop23d 41884 . . 3 (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))))
7 simpl 483 . . . 4 ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → 𝒫 𝐴𝑤)
87reximi 3178 . . 3 (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → ∃𝑤𝑈 𝒫 𝐴𝑤)
96, 8syl 17 . 2 (𝜑 → ∃𝑤𝑈 𝒫 𝐴𝑤)
101, 2, 9mnuss2d 41882 1 (𝜑 → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wrex 3065  Vcvv 3432  wss 3887  c0 4256  𝒫 cpw 4533   cuni 4839
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-uni 4840
This theorem is referenced by:  mnusnd  41886  mnuprssd  41887  mnugrud  41902
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