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Theorem grumnud 42658
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 2733 . 2 ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺))
4 brxp 5685 . . . 4 (𝑖(𝐺 × 𝐺) ↔ (𝑖𝐺𝐺))
5 brin 5161 . . . . 5 (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) ↔ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}𝑖(𝐺 × 𝐺)))
65rbaib 540 . . . 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 3451 . . . 4 𝑖 ∈ V
9 vex 3451 . . . 4 ∈ V
10 simpr 486 . . . . . . . 8 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
1110unieqd 4883 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
12 simplr 768 . . . . . . 7 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → 𝑐 = )
1311, 12eqeq12d 2749 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → ( 𝑑 = 𝑐 𝑗 = ))
14 elequ1 2114 . . . . . . 7 (𝑑 = 𝑗 → (𝑑𝑓𝑗𝑓))
1514adantl 483 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑑𝑓𝑗𝑓))
16 eleq12 2824 . . . . . . 7 ((𝑏 = 𝑖𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1716adantlr 714 . . . . . 6 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (𝑏𝑑𝑖𝑗))
1813, 15, 173anbi123d 1437 . . . . 5 (((𝑏 = 𝑖𝑐 = ) ∧ 𝑑 = 𝑗) → (( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ( 𝑗 = 𝑗𝑓𝑖𝑗)))
1918cbvexdvaw 2043 . . . 4 ((𝑏 = 𝑖𝑐 = ) → (∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑) ↔ ∃𝑗( 𝑗 = 𝑗𝑓𝑖𝑗)))
20 eqid 2733 . . . 4 {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)}
218, 9, 19, 20braba 5498 . . 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 1218 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑗)
24 simpr 486 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
2523, 24eleqtrrd 2837 . . . 4 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑖𝑢)
2624unieqd 4883 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
27 simplr1 1216 . . . . . 6 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑗 = )
2826, 27eqtrd 2773 . . . . 5 ((( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = )
29 simpll 766 . . . . 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 2834 . . . 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 513 . . 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 1196 . . 3 (( ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑( 𝑑 = 𝑐𝑑𝑓𝑏𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ ( 𝑗 = 𝑗𝑓𝑖𝑗)) → 𝑗𝑓)
3331, 32rspcime 3586 . 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 42657 1 (𝜑𝐺𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  cin 3913  wss 3914  𝒫 cpw 4564   cuni 4869   class class class wbr 5109  {copab 5171   × cxp 5635  Univcgru 10734   Coll ccoll 42622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585  ax-ac2 10407
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-tc 9681  df-r1 9708  df-rank 9709  df-card 9883  df-cf 9885  df-acn 9886  df-ac 10060  df-wina 10628  df-ina 10629  df-gru 10735  df-scott 42608  df-coll 42623
This theorem is referenced by:  grumnueq  42659
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