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Theorem mnuprd 42498
Description: Minimal universes are closed under pairing. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuprd.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuprd.2 (𝜑𝑈𝑀)
mnuprd.3 (𝜑𝐴𝑈)
mnuprd.4 (𝜑𝐵𝑈)
Assertion
Ref Expression
mnuprd (𝜑 → {𝐴, 𝐵} ∈ 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 mnuprd
StepHypRef Expression
1 mnuprd.1 . . 3 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnuprd.2 . . . 4 (𝜑𝑈𝑀)
32adantr 481 . . 3 ((𝜑𝐴 = ∅) → 𝑈𝑀)
4 mnuprd.4 . . . 4 (𝜑𝐵𝑈)
54adantr 481 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝑈)
6 simpr 485 . . . 4 ((𝜑𝐴 = ∅) → 𝐴 = ∅)
7 0ss 4354 . . . 4 ∅ ⊆ 𝐵
86, 7eqsstrdi 3996 . . 3 ((𝜑𝐴 = ∅) → 𝐴𝐵)
9 ssidd 3965 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝐵)
101, 3, 5, 8, 9mnuprssd 42491 . 2 ((𝜑𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11 eqid 2736 . . 3 {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}}
122adantr 481 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈𝑀)
13 mnuprd.3 . . . 4 (𝜑𝐴𝑈)
1413adantr 481 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴𝑈)
154adantr 481 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵𝑈)
16 simpr 485 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅)
171, 11, 12, 14, 15, 16mnuprdlem4 42497 . 2 ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
1810, 17pm2.61dan 811 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1539   = wceq 1541  wcel 2106  {cab 2713  wral 3062  wrex 3071  wss 3908  c0 4280  𝒫 cpw 4558  {csn 4584  {cpr 4586   cuni 4863
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-pw 4560  df-sn 4585  df-pr 4587  df-uni 4864
This theorem is referenced by:  mnuund  42500  mnurndlem2  42504  mnugrud  42506
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