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Theorem mnuop3d 40903
 Description: Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuop3d.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuop3d.2 (𝜑𝑈𝑀)
mnuop3d.3 (𝜑𝐴𝑈)
mnuop3d.4 (𝜑𝐹𝑈)
Assertion
Ref Expression
mnuop3d (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))
Distinct variable groups:   𝑣,𝐹   𝑤,𝐴,𝑖   𝜑,𝑤,𝑣,𝑖   𝑤,𝑢,𝐹,𝑖   𝑤,𝑈,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙,𝑣,𝑖   𝑤,𝑟   𝑢,𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙,𝑖
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑣,𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑤,𝑣,𝑢,𝑖,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuop3d
StepHypRef Expression
1 mnuop3d.1 . . 3 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnuop3d.2 . . 3 (𝜑𝑈𝑀)
3 mnuop3d.3 . . 3 (𝜑𝐴𝑈)
4 mnuop3d.4 . . . 4 (𝜑𝐹𝑈)
52, 4sselpwd 5216 . . 3 (𝜑𝐹 ∈ 𝒫 𝑈)
61, 2, 3, 5mnuop23d 40898 . 2 (𝜑 → ∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
74sseld 3952 . . . . . . . . 9 (𝜑 → (𝑣𝐹𝑣𝑈))
87adantrd 495 . . . . . . . 8 (𝜑 → ((𝑣𝐹𝑖𝑣) → 𝑣𝑈))
9 pm3.22 463 . . . . . . . 8 ((𝑣𝐹𝑖𝑣) → (𝑖𝑣𝑣𝐹))
108, 9jca2 517 . . . . . . 7 (𝜑 → ((𝑣𝐹𝑖𝑣) → (𝑣𝑈 ∧ (𝑖𝑣𝑣𝐹))))
1110reximdv2 3263 . . . . . 6 (𝜑 → (∃𝑣𝐹 𝑖𝑣 → ∃𝑣𝑈 (𝑖𝑣𝑣𝐹)))
1211imim1d 82 . . . . 5 (𝜑 → ((∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)) → (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1312ralimdv 3173 . . . 4 (𝜑 → (∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)) → ∀𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1413adantld 494 . . 3 (𝜑 → ((𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))) → ∀𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
1514reximdv 3265 . 2 (𝜑 → (∃𝑤𝑈 (𝒫 𝐴𝑤 ∧ ∀𝑖𝐴 (∃𝑣𝑈 (𝑖𝑣𝑣𝐹) → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))) → ∃𝑤𝑈𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤))))
166, 15mpd 15 1 (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣𝐹 𝑖𝑣 → ∃𝑢𝐹 (𝑖𝑢 𝑢𝑤)))
 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  ∪ 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 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-in 3926  df-ss 3936  df-pw 4524  df-uni 4825 This theorem is referenced by:  mnuprdlem4  40907  mnuunid  40909  mnurndlem2  40914
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