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Theorem mnu0eld 40893
 Description: A nonempty minimal universe contains the empty set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnu0eld.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnu0eld.2 (𝜑𝑈𝑀)
mnu0eld.3 (𝜑𝐴𝑈)
Assertion
Ref Expression
mnu0eld (𝜑 → ∅ ∈ 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 mnu0eld
StepHypRef Expression
1 mnu0eld.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnu0eld.2 . 2 (𝜑𝑈𝑀)
3 mnu0eld.3 . 2 (𝜑𝐴𝑈)
4 0ss 4333 . . 3 ∅ ⊆ 𝐴
54a1i 11 . 2 (𝜑 → ∅ ⊆ 𝐴)
61, 2, 3, 5mnussd 40891 1 (𝜑 → ∅ ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2115  {cab 2802  ∀wral 3133  ∃wrex 3134   ⊆ wss 3919  ∅c0 4276  𝒫 cpw 4522  ∪ cuni 4824 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-uni 4825 This theorem is referenced by:  mnuprdlem4  40903
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