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Theorem mnuund 44297
Description: Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuund.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuund.2 (𝜑𝑈𝑀)
mnuund.3 (𝜑𝐴𝑈)
mnuund.4 (𝜑𝐵𝑈)
Assertion
Ref Expression
mnuund (𝜑 → (𝐴𝐵) ∈ 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)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuund
StepHypRef Expression
1 mnuund.3 . . 3 (𝜑𝐴𝑈)
2 mnuund.4 . . 3 (𝜑𝐵𝑈)
3 uniprg 4923 . . 3 ((𝐴𝑈𝐵𝑈) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3syl2anc 584 . 2 (𝜑 {𝐴, 𝐵} = (𝐴𝐵))
5 mnuund.1 . . 3 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
6 mnuund.2 . . 3 (𝜑𝑈𝑀)
75, 6, 1, 2mnuprd 44295 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
85, 6, 7mnuunid 44296 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
94, 8eqeltrrd 2842 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  cun 3949  wss 3951  𝒫 cpw 4600  {cpr 4628   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908
This theorem is referenced by: (None)
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