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Theorem mnuprss2d 44179
Description: Special case of mnuprssd 44178. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuprss2d.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuprss2d.2 (𝜑𝑈𝑀)
mnuprss2d.3 (𝜑𝐶𝑈)
mnuprss2d.4 𝐴𝐶
mnuprss2d.5 𝐵𝐶
Assertion
Ref Expression
mnuprss2d (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 mnuprss2d
StepHypRef Expression
1 mnuprss2d.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnuprss2d.2 . 2 (𝜑𝑈𝑀)
3 mnuprss2d.3 . 2 (𝜑𝐶𝑈)
4 mnuprss2d.4 . . 3 𝐴𝐶
54a1i 11 . 2 (𝜑𝐴𝐶)
6 mnuprss2d.5 . . 3 𝐵𝐶
76a1i 11 . 2 (𝜑𝐵𝐶)
81, 2, 3, 5, 7mnuprssd 44178 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2103  {cab 2711  wral 3063  wrex 3072  wss 3970  𝒫 cpw 4622  {cpr 4650   cuni 4931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932
This theorem is referenced by:  mnuprdlem4  44184
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