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Theorem mnusnd 41395
 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 41394 . 2 (𝜑 → 𝒫 𝐴𝑈)
5 snsspw 4736 . . 3 {𝐴} ⊆ 𝒫 𝐴
65a1i 11 . 2 (𝜑 → {𝐴} ⊆ 𝒫 𝐴)
71, 2, 4, 6mnussd 41390 1 (𝜑 → {𝐴} ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1537   = wceq 1539   ∈ wcel 2112  {cab 2736  ∀wral 3071  ∃wrex 3072   ⊆ wss 3861  𝒫 cpw 4498  {csn 4526  ∪ cuni 4802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3864  df-in 3868  df-ss 3878  df-nul 4229  df-pw 4500  df-sn 4527  df-uni 4803 This theorem is referenced by:  mnuprdlem4  41402
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