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Theorem mnuprd 44272
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 480 . . 3 ((𝜑𝐴 = ∅) → 𝑈𝑀)
4 mnuprd.4 . . . 4 (𝜑𝐵𝑈)
54adantr 480 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝑈)
6 simpr 484 . . . 4 ((𝜑𝐴 = ∅) → 𝐴 = ∅)
7 0ss 4406 . . . 4 ∅ ⊆ 𝐵
86, 7eqsstrdi 4050 . . 3 ((𝜑𝐴 = ∅) → 𝐴𝐵)
9 ssidd 4019 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝐵)
101, 3, 5, 8, 9mnuprssd 44265 . 2 ((𝜑𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11 eqid 2735 . . 3 {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}}
122adantr 480 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈𝑀)
13 mnuprd.3 . . . 4 (𝜑𝐴𝑈)
1413adantr 480 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴𝑈)
154adantr 480 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵𝑈)
16 simpr 484 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅)
171, 11, 12, 14, 15, 16mnuprdlem4 44271 . 2 ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
1810, 17pm2.61dan 813 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by:  mnuund  44274  mnurndlem2  44278  mnugrud  44280
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