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Theorem mnupwd 42639
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 5268 . . . . 5 ∅ ∈ V
54a1i 11 . . . 4 (𝜑 → ∅ ∈ V)
61, 2, 3, 5mnuop23d 42638 . . 3 (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))))
7 simpl 484 . . . 4 ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → 𝒫 𝐴𝑤)
87reximi 3084 . . 3 (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖𝑢 𝑢𝑤))) → ∃𝑤𝑈 𝒫 𝐴𝑤)
96, 8syl 17 . 2 (𝜑 → ∃𝑤𝑈 𝒫 𝐴𝑤)
101, 2, 9mnuss2d 42636 1 (𝜑 → 𝒫 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3447  wss 3914  c0 4286  𝒫 cpw 4564   cuni 4869
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-nul 4287  df-pw 4566  df-uni 4870
This theorem is referenced by:  mnusnd  42640  mnuprssd  42641  mnugrud  42656
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