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Theorem mnupwd 41468
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 5185 . . . . 5 ∅ ∈ V
54a1i 11 . . . 4 (𝜑 → ∅ ∈ V)
61, 2, 3, 5mnuop23d 41467 . . 3 (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))))
7 simpl 486 . . . 4 ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → 𝒫 𝐴𝑤)
87reximi 3158 . . 3 (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → ∃𝑤𝑈 𝒫 𝐴𝑤)
96, 8syl 17 . 2 (𝜑 → ∃𝑤𝑈 𝒫 𝐴𝑤)
101, 2, 9mnuss2d 41465 1 (𝜑 → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1540   = wceq 1542  wcel 2114  {cab 2717  wral 3054  wrex 3055  Vcvv 3400  wss 3853  c0 4221  𝒫 cpw 4498   cuni 4806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-in 3860  df-ss 3870  df-nul 4222  df-pw 4500  df-uni 4807
This theorem is referenced by:  mnusnd  41469  mnuprssd  41470  mnugrud  41485
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