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Theorem mnusnd 43979
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 43978 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 snsspw 4843 . . 3 {𝐴} ⊆ 𝒫 𝐴
65a1i 11 . 2 (𝜑 → {𝐴} ⊆ 𝒫 𝐴)
71, 2, 4, 6mnussd 43974 1 (𝜑 → {𝐴} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1532   = wceq 1534  wcel 2099  {cab 2703  wral 3051  wrex 3060  wss 3946  𝒫 cpw 4597  {csn 4623   cuni 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-in 3953  df-ss 3963  df-nul 4323  df-pw 4599  df-sn 4624  df-uni 4906
This theorem is referenced by:  mnuprdlem4  43986
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