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Theorem mnuund 42217
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 4869 . . 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 42215 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
85, 6, 7mnuunid 42216 . 2 (𝜑 {𝐴, 𝐵} ∈ 𝑈)
94, 8eqeltrrd 2838 1 (𝜑 → (𝐴𝐵) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1538   = wceq 1540  wcel 2105  {cab 2713  wral 3061  wrex 3070  cun 3896  wss 3898  𝒫 cpw 4547  {cpr 4575   cuni 4852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-pw 4549  df-sn 4574  df-pr 4576  df-uni 4853
This theorem is referenced by: (None)
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