Theorem grumnud 40671

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 2821	. 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 5601	. . . 4 ⊢ (𝑖(𝐺 × 𝐺)ℎ ↔ (𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺))
5		brin 5118	. . . . 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 541	. . . 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 237	. . 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 3497	. . . 4 ⊢ 𝑖 ∈ V
9		vex 3497	. . . 4 ⊢ ℎ ∈ V
10		simpr 487	. . . . . . . 8 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
11	10	unieqd 4852	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ∪ 𝑑 = ∪ 𝑗)
12		simplr 767	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑐 = ℎ)
13	11, 12	eqeq12d 2837	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (∪ 𝑑 = 𝑐 ↔ ∪ 𝑗 = ℎ))
14		elequ1 2121	. . . . . . 7 ⊢ (𝑑 = 𝑗 → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
15	14	adantl 484	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
16		eleq12 2902	. . . . . . 7 ⊢ ((𝑏 = 𝑖 ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
17	16	adantlr 713	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
18	13, 15, 17	3anbi123d 1432	. . . . 5 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ((∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
19	18	cbvexdvaw 2046	. . . 4 ⊢ ((𝑏 = 𝑖 ∧ 𝑐 = ℎ) → (∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
20		eqid 2821	. . . 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 5424	. . 3 ⊢ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
22	7, 21	syl6bb 289	. 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 1213	. . . . 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 487	. . . . 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 2916	. . . 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 4852	. . . . . 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 1211	. . . . . 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 2856	. . . . 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 765	. . . . 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 2913	. . . 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 514	. . 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 1191	. . 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 3627	. 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 40670	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838   class class class wbr 5066  {copab 5128   × cxp 5553  Univcgru 10212   Coll ccoll 40635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056  ax-inf2 9104  ax-ac2 9885

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-tc 9179  df-r1 9193  df-rank 9194  df-card 9368  df-cf 9370  df-acn 9371  df-ac 9542  df-wina 10106  df-ina 10107  df-gru 10213  df-scott 40621  df-coll 40636

This theorem is referenced by:  grumnueq  40672