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Theorem mnusnd 40127
Description: Minimal universes are closed under singletons. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnusnd.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnusnd.2 (𝜑𝑈𝑀)
mnusnd.3 (𝜑𝐴𝑈)
Assertion
Ref Expression
mnusnd (𝜑 → {𝐴} ∈ 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)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnusnd
StepHypRef Expression
1 mnusnd.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnusnd.2 . 2 (𝜑𝑈𝑀)
3 mnusnd.3 . . 3 (𝜑𝐴𝑈)
41, 2, 3mnupwd 40126 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 snsspw 4686 . . 3 {𝐴} ⊆ 𝒫 𝐴
65a1i 11 . 2 (𝜑 → {𝐴} ⊆ 𝒫 𝐴)
71, 2, 4, 6mnussd 40122 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1520   = wceq 1522  wcel 2081  {cab 2775  wral 3105  wrex 3106  wss 3863  𝒫 cpw 4457  {csn 4476   cuni 4749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439  df-dif 3866  df-in 3870  df-ss 3878  df-nul 4216  df-pw 4459  df-sn 4477  df-uni 4750
This theorem is referenced by:  mnuprdlem4  40134
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