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Theorem mnuprssd 40897
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 40895 . 2 (𝜑 → 𝒫 𝐶𝑈)
5 mnuprssd.4 . . . 4 (𝜑𝐴𝐶)
63, 5sselpwd 5216 . . 3 (𝜑𝐴 ∈ 𝒫 𝐶)
7 mnuprssd.5 . . . 4 (𝜑𝐵𝐶)
83, 7sselpwd 5216 . . 3 (𝜑𝐵 ∈ 𝒫 𝐶)
96, 8prssd 4739 . 2 (𝜑 → {𝐴, 𝐵} ⊆ 𝒫 𝐶)
101, 2, 4, 9mnussd 40891 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wcel 2115  {cab 2802  wral 3133  wrex 3134  wss 3919  𝒫 cpw 4522  {cpr 4552   cuni 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553  df-uni 4825
This theorem is referenced by:  mnuprss2d  40898  mnuprd  40904
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