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Theorem mnuprd 44295
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 4400 . . . 4 ∅ ⊆ 𝐵
86, 7eqsstrdi 4028 . . 3 ((𝜑𝐴 = ∅) → 𝐴𝐵)
9 ssidd 4007 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝐵)
101, 3, 5, 8, 9mnuprssd 44288 . 2 ((𝜑𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11 eqid 2737 . . 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 44294 . 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 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  {cpr 4628   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908
This theorem is referenced by:  mnuund  44297  mnurndlem2  44301  mnugrud  44303
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