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 40140
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 2797 . 2 ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))
4 brxp 5496 . . . 4 (𝑖(𝐺 × 𝐺) ↔ (𝑖𝐺𝐺))
5 brin 5020 . . . . 5 (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}𝑖(𝐺 × 𝐺)))
65rbaib 539 . . . 4 (𝑖(𝐺 × 𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}))
74, 6sylbir 236 . . 3 ((𝑖𝐺𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}))
8 vex 3443 . . . 4 𝑖 ∈ V
9 vex 3443 . . . 4 ∈ V
10 simpr 485 . . . . . . . 8 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
1110unieqd 4761 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
12 simplr 765 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑐 = )
1311, 12eqeq12d 2812 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → ( 𝑑 = 𝑐 𝑗 = ))
14 elequ1 2090 . . . . . . 7 (𝑑 = 𝑗 → (𝑑𝑓𝑗𝑓))
1514adantl 482 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑑𝑓𝑗𝑓))
16 eleq12 2874 . . . . . . 7 ((𝑏 = 𝑖𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1716adantlr 711 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1813, 15, 173anbi123d 1428 . . . . 5 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ( 𝑗 = 𝑗𝑓𝑖𝑗)))
1918cbvexdva 2389 . . . 4 ((𝑏 = 𝑖𝑐 = ) → (∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
20 eqid 2797 . . . 4 {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}
218, 9, 19, 20braba 5321 . . 3 (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗))
227, 21syl6bb 288 . 2 ((𝑖𝐺𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
23 simplr3 1210 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑗)
24 simpr 485 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
2523, 24eleqtrrd 2888 . . . 4 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑢)
2624unieqd 4761 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
27 simplr1 1208 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑗 = )
2826, 27eqtrd 2833 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = )
29 simpll 763 . . . . 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 2885 . . . 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 512 . . 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 1188 . . 3 (( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → 𝑗𝑓)
3331, 32rr-rspce 40066 . 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 40139 1 (𝜑𝐺𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080  wal 1523   = wceq 1525  wex 1765  wcel 2083  {cab 2777  wral 3107  wrex 3108  cin 3864  wss 3865  𝒫 cpw 4459   cuni 4751   class class class wbr 4968  {copab 5030   × cxp 5448  Univcgru 10065   Coll ccoll 40104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-reg 8909  ax-inf2 8957  ax-ac2 9738
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-tc 9032  df-r1 9046  df-rank 9047  df-card 9221  df-cf 9223  df-acn 9224  df-ac 9395  df-wina 9959  df-ina 9960  df-gru 10066  df-scott 40090  df-coll 40105
This theorem is referenced by:  grumnueq  40141
  Copyright terms: Public domain W3C validator