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Theorem mnuprssd 42641
Description: A minimal universe contains pairs of subsets of an element of the universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuprssd.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuprssd.2 (𝜑𝑈𝑀)
mnuprssd.3 (𝜑𝐶𝑈)
mnuprssd.4 (𝜑𝐴𝐶)
mnuprssd.5 (𝜑𝐵𝐶)
Assertion
Ref Expression
mnuprssd (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,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)   𝐶(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuprssd
StepHypRef Expression
1 mnuprssd.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnuprssd.2 . 2 (𝜑𝑈𝑀)
3 mnuprssd.3 . . 3 (𝜑𝐶𝑈)
41, 2, 3mnupwd 42639 . 2 (𝜑 → 𝒫 𝐶𝑈)
5 mnuprssd.4 . . . 4 (𝜑𝐴𝐶)
63, 5sselpwd 5287 . . 3 (𝜑𝐴 ∈ 𝒫 𝐶)
7 mnuprssd.5 . . . 4 (𝜑𝐵𝐶)
83, 7sselpwd 5287 . . 3 (𝜑𝐵 ∈ 𝒫 𝐶)
96, 8prssd 4786 . 2 (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝐶)
101, 2, 4, 9mnussd 42635 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  wss 3914  𝒫 cpw 4564  {cpr 4592   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-or 847  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-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-pw 4566  df-sn 4591  df-pr 4593  df-uni 4870
This theorem is referenced by:  mnuprss2d  42642  mnuprd  42648
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