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Theorem grumnud 44282
Description: Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.)
Hypotheses
Ref Expression
grumnud.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
grumnud.2 (𝜑𝐺 ∈ Univ)
Assertion
Ref Expression
grumnud (𝜑𝐺𝑀)
Distinct variable groups:   𝑘,𝑚,𝑛,𝐺,𝑞,𝑝,𝑙   𝑘,𝑟,𝑚,𝑛,𝐺,𝑝,𝑙
Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem grumnud
Dummy variables 𝑧 𝑓 𝑖 𝑗 𝑢 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grumnud.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 grumnud.2 . 2 (𝜑𝐺 ∈ Univ)
3 eqid 2735 . 2 ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))
4 brxp 5738 . . . 4 (𝑖(𝐺 × 𝐺) ↔ (𝑖𝐺𝐺))
5 brin 5200 . . . . 5 (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}𝑖(𝐺 × 𝐺)))
65rbaib 538 . . . 4 (𝑖(𝐺 × 𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}))
74, 6sylbir 235 . . 3 ((𝑖𝐺𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}))
8 vex 3482 . . . 4 𝑖 ∈ V
9 vex 3482 . . . 4 ∈ V
10 simpr 484 . . . . . . . 8 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
1110unieqd 4925 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
12 simplr 769 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑐 = )
1311, 12eqeq12d 2751 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → ( 𝑑 = 𝑐 𝑗 = ))
14 elequ1 2113 . . . . . . 7 (𝑑 = 𝑗 → (𝑑𝑓𝑗𝑓))
1514adantl 481 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑑𝑓𝑗𝑓))
16 eleq12 2829 . . . . . . 7 ((𝑏 = 𝑖𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1716adantlr 715 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1813, 15, 173anbi123d 1435 . . . . 5 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ( 𝑗 = 𝑗𝑓𝑖𝑗)))
1918cbvexdvaw 2036 . . . 4 ((𝑏 = 𝑖𝑐 = ) → (∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
20 eqid 2735 . . . 4 {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}
218, 9, 19, 20braba 5547 . . 3 (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗))
227, 21bitrdi 287 . 2 ((𝑖𝐺𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
23 simplr3 1216 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑗)
24 simpr 484 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
2523, 24eleqtrrd 2842 . . . 4 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑢)
2624unieqd 4925 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
27 simplr1 1214 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑗 = )
2826, 27eqtrd 2775 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = )
29 simpll 767 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧))
3028, 29eqeltrd 2839 . . . 4 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧))
3125, 30jca 511 . . 3 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → (𝑖𝑢 𝑢 ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧)))
32 simpr2 1194 . . 3 (( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → 𝑗𝑓)
3331, 32rspcime 3627 . 2 (( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → ∃𝑢𝑓 (𝑖𝑢 𝑢 ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧)))
341, 2, 3, 22, 33grumnudlem 44281 1 (𝜑𝐺𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  {copab 5210   × cxp 5687  Univcgru 10828   Coll ccoll 44246
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-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-tc 9775  df-r1 9802  df-rank 9803  df-card 9977  df-cf 9979  df-acn 9980  df-ac 10154  df-wina 10722  df-ina 10723  df-gru 10829  df-scott 44232  df-coll 44247
This theorem is referenced by:  grumnueq  44283
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