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

Theorem grumnud 43726
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 2727 . 2 ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))
4 brxp 5729 . . . 4 (𝑖(𝐺 × 𝐺) ↔ (𝑖𝐺𝐺))
5 brin 5202 . . . . 5 (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}𝑖(𝐺 × 𝐺)))
65rbaib 537 . . . 4 (𝑖(𝐺 × 𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}))
74, 6sylbir 234 . . 3 ((𝑖𝐺𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}))
8 vex 3475 . . . 4 𝑖 ∈ V
9 vex 3475 . . . 4 ∈ V
10 simpr 483 . . . . . . . 8 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
1110unieqd 4923 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
12 simplr 767 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑐 = )
1311, 12eqeq12d 2743 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → ( 𝑑 = 𝑐 𝑗 = ))
14 elequ1 2105 . . . . . . 7 (𝑑 = 𝑗 → (𝑑𝑓𝑗𝑓))
1514adantl 480 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑑𝑓𝑗𝑓))
16 eleq12 2818 . . . . . . 7 ((𝑏 = 𝑖𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1716adantlr 713 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1813, 15, 173anbi123d 1432 . . . . 5 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ( 𝑗 = 𝑗𝑓𝑖𝑗)))
1918cbvexdvaw 2034 . . . 4 ((𝑏 = 𝑖𝑐 = ) → (∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
20 eqid 2727 . . . 4 {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}
218, 9, 19, 20braba 5541 . . 3 (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗))
227, 21bitrdi 286 . 2 ((𝑖𝐺𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
23 simplr3 1214 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑗)
24 simpr 483 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
2523, 24eleqtrrd 2831 . . . 4 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑢)
2624unieqd 4923 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
27 simplr1 1212 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑗 = )
2826, 27eqtrd 2767 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = )
29 simpll 765 . . . . 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 2828 . . . 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 510 . . 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 1192 . . 3 (( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → 𝑗𝑓)
3331, 32rspcime 3614 . 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 43725 1 (𝜑𝐺𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2704  wral 3057  wrex 3066  cin 3946  wss 3947  𝒫 cpw 4604   cuni 4910   class class class wbr 5150  {copab 5212   × cxp 5678  Univcgru 10819   Coll ccoll 43690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-reg 9621  ax-inf2 9670  ax-ac2 10492
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-iin 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-se 5636  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-isom 6560  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-tc 9766  df-r1 9793  df-rank 9794  df-card 9968  df-cf 9970  df-acn 9971  df-ac 10145  df-wina 10713  df-ina 10714  df-gru 10820  df-scott 43676  df-coll 43691
This theorem is referenced by:  grumnueq  43727
  Copyright terms: Public domain W3C validator