Mathbox for Rohan Ridenour < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mnuprd Structured version   Visualization version   GIF version

Theorem mnuprd 41385
 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 484 . . 3 ((𝜑𝐴 = ∅) → 𝑈𝑀)
4 mnuprd.4 . . . 4 (𝜑𝐵𝑈)
54adantr 484 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝑈)
6 simpr 488 . . . 4 ((𝜑𝐴 = ∅) → 𝐴 = ∅)
7 0ss 4295 . . . 4 ∅ ⊆ 𝐵
86, 7eqsstrdi 3948 . . 3 ((𝜑𝐴 = ∅) → 𝐴𝐵)
9 ssidd 3917 . . 3 ((𝜑𝐴 = ∅) → 𝐵𝐵)
101, 3, 5, 8, 9mnuprssd 41378 . 2 ((𝜑𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
11 eqid 2758 . . 3 {{∅, {𝐴}}, {{∅}, {𝐵}}} = {{∅, {𝐴}}, {{∅}, {𝐵}}}
122adantr 484 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑈𝑀)
13 mnuprd.3 . . . 4 (𝜑𝐴𝑈)
1413adantr 484 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴𝑈)
154adantr 484 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐵𝑈)
16 simpr 488 . . 3 ((𝜑 ∧ ¬ 𝐴 = ∅) → ¬ 𝐴 = ∅)
171, 11, 12, 14, 15, 16mnuprdlem4 41384 . 2 ((𝜑 ∧ ¬ 𝐴 = ∅) → {𝐴, 𝐵} ∈ 𝑈)
1810, 17pm2.61dan 812 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {csn 4525  {cpr 4527  ∪ cuni 4801 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499  df-sn 4526  df-pr 4528  df-uni 4802 This theorem is referenced by:  mnuund  41387  mnurndlem2  41391  mnugrud  41393
 Copyright terms: Public domain W3C validator