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Theorem mnuprssd 44265
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 44263 . 2 (𝜑 → 𝒫 𝐶𝑈)
5 mnuprssd.4 . . . 4 (𝜑𝐴𝐶)
63, 5sselpwd 5334 . . 3 (𝜑𝐴 ∈ 𝒫 𝐶)
7 mnuprssd.5 . . . 4 (𝜑𝐵𝐶)
83, 7sselpwd 5334 . . 3 (𝜑𝐵 ∈ 𝒫 𝐶)
96, 8prssd 4827 . 2 (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝐶)
101, 2, 4, 9mnussd 44259 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  𝒫 cpw 4605  {cpr 4633   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by:  mnuprss2d  44266  mnuprd  44272
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