Theorem grumnud 45462

Description: Grothendieck universes are minimal universes. (Contributed by Rohan Ridenour, 12-Aug-2023.)

Hypotheses

Ref	Expression
grumnud.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
grumnud.2	⊢ (𝜑 → 𝐺 ∈ Univ)

Assertion

Ref	Expression
grumnud	⊢ (𝜑 → 𝐺 ∈ 𝑀)

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 ℎ 𝑖 𝑗 𝑢 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grumnud.1	. 2 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
2		grumnud.2	. 2 ⊢ (𝜑 → 𝐺 ∈ Univ)
3		eqid 2763	. 2 ⊢ ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) = ({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))
4		brxp 5704	. . . 4 ⊢ (𝑖(𝐺 × 𝐺)ℎ ↔ (𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺))
5		brin 5160	. . . . 5 ⊢ (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))ℎ ↔ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}ℎ ∧ 𝑖(𝐺 × 𝐺)ℎ))
6	5	rbaib 548	. . . 4 ⊢ (𝑖(𝐺 × 𝐺)ℎ → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))ℎ ↔ 𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}ℎ))
7	4, 6	sylbir 238	. . 3 ⊢ ((𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))ℎ ↔ 𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}ℎ))
8		vex 3458	. . . 4 ⊢ 𝑖 ∈ V
9		vex 3458	. . . 4 ⊢ ℎ ∈ V
10		simpr 490	. . . . . . . 8 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
11	10	unieqd 4883	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ∪ 𝑑 = ∪ 𝑗)
12		simplr 781	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑐 = ℎ)
13	11, 12	eqeq12d 2779	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (∪ 𝑑 = 𝑐 ↔ ∪ 𝑗 = ℎ))
14		elequ1 2154	. . . . . . 7 ⊢ (𝑑 = 𝑗 → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
15	14	adantl 487	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
16		eleq12 2854	. . . . . . 7 ⊢ ((𝑏 = 𝑖 ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
17	16	adantlr 728	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
18	13, 15, 17	3anbi123d 1465	. . . . 5 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ((∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
19	18	cbvexdvaw 2074	. . . 4 ⊢ ((𝑏 = 𝑖 ∧ 𝑐 = ℎ) → (∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
20		eqid 2763	. . . 4 ⊢ {⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} = {⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}
21	8, 9, 19, 20	braba 5515	. . 3 ⊢ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
22	7, 21	bitrdi 290	. 2 ⊢ ((𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺) → (𝑖({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺))ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
23		simplr3 1236	. . . . 5 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → 𝑖 ∈ 𝑗)
24		simpr 490	. . . . 5 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → 𝑢 = 𝑗)
25	23, 24	eleqtrrd 2867	. . . 4 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → 𝑖 ∈ 𝑢)
26	24	unieqd 4883	. . . . . 6 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → ∪ 𝑢 = ∪ 𝑗)
27		simplr1 1234	. . . . . 6 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → ∪ 𝑗 = ℎ)
28	26, 27	eqtrd 2798	. . . . 5 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → ∪ 𝑢 = ℎ)
29		simpll 779	. . . . 5 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧))
30	28, 29	eqeltrd 2864	. . . 4 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → ∪ 𝑢 ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧))
31	25, 30	jca 521	. . 3 ⊢ (((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) ∧ 𝑢 = 𝑗) → (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧)))
32		simpr2 1214	. . 3 ⊢ ((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) → 𝑗 ∈ 𝑓)
33	31, 32	rspcime 3585	. 2 ⊢ ((ℎ ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧) ∧ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ∈ (({⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)} ∩ (𝐺 × 𝐺)) Coll 𝑧)))
34	1, 2, 3, 22, 33	grumnudlem 45461	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  {copab 5170   × cxp 5653  Univcgru 10902   Coll ccoll 45426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-reg 9613  ax-inf2 9669  ax-ac2 10568

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-tc 9763  df-r1 9795  df-rank 9796  df-scott 9956  df-card 10047  df-cf 10049  df-acn 10050  df-ac 10222  df-wina 10796  df-ina 10797  df-gru 10903  df-coll 45427

This theorem is used by:  grumnueq  45463