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Theorem grumnud 43865
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 2725 . 2 ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))
4 brxp 5727 . . . 4 (𝑖(𝐺 × 𝐺) ↔ (𝑖𝐺𝐺))
5 brin 5201 . . . . 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 3465 . . . 4 𝑖 ∈ V
9 vex 3465 . . . 4 ∈ V
10 simpr 483 . . . . . . . 8 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
1110unieqd 4922 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
12 simplr 767 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑐 = )
1311, 12eqeq12d 2741 . . . . . 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 2815 . . . . . . 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 2725 . . . 4 {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}
218, 9, 19, 20braba 5539 . . 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 2828 . . . 4 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑢)
2624unieqd 4922 . . . . . 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 2765 . . . . 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 2825 . . . 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 3611 . 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 43864 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 2702  wral 3050  wrex 3059  cin 3943  wss 3944  𝒫 cpw 4604   cuni 4909   class class class wbr 5149  {copab 5211   × cxp 5676  Univcgru 10815   Coll ccoll 43829
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-reg 9617  ax-inf2 9666  ax-ac2 10488
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-tc 9762  df-r1 9789  df-rank 9790  df-card 9964  df-cf 9966  df-acn 9967  df-ac 10141  df-wina 10709  df-ina 10710  df-gru 10816  df-scott 43815  df-coll 43830
This theorem is referenced by:  grumnueq  43866
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