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Theorem mnupwd 44238
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 5325 . . . . 5 ∅ ∈ V
54a1i 11 . . . 4 (𝜑 → ∅ ∈ V)
61, 2, 3, 5mnuop23d 44237 . . 3 (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))))
7 simpl 482 . . . 4 ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → 𝒫 𝐴𝑤)
87reximi 3090 . . 3 (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → ∃𝑤𝑈 𝒫 𝐴𝑤)
96, 8syl 17 . 2 (𝜑 → ∃𝑤𝑈 𝒫 𝐴𝑤)
101, 2, 9mnuss2d 44235 1 (𝜑 → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2108  {cab 2717  wral 3067  wrex 3076  Vcvv 3488  wss 3976  c0 4352  𝒫 cpw 4622   cuni 4931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-nul 4353  df-pw 4624  df-uni 4932
This theorem is referenced by:  mnusnd  44239  mnuprssd  44240  mnugrud  44255
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