Theorem grumnud 44282

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 2730	. 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 5690	. . . 4 ⊢ (𝑖(𝐺 × 𝐺)ℎ ↔ (𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺))
5		brin 5162	. . . . 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 538	. . . 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 235	. . 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 3454	. . . 4 ⊢ 𝑖 ∈ V
9		vex 3454	. . . 4 ⊢ ℎ ∈ V
10		simpr 484	. . . . . . . 8 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
11	10	unieqd 4887	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ∪ 𝑑 = ∪ 𝑗)
12		simplr 768	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑐 = ℎ)
13	11, 12	eqeq12d 2746	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (∪ 𝑑 = 𝑐 ↔ ∪ 𝑗 = ℎ))
14		elequ1 2116	. . . . . . 7 ⊢ (𝑑 = 𝑗 → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
15	14	adantl 481	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
16		eleq12 2819	. . . . . . 7 ⊢ ((𝑏 = 𝑖 ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
17	16	adantlr 715	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
18	13, 15, 17	3anbi123d 1438	. . . . 5 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ((∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
19	18	cbvexdvaw 2039	. . . 4 ⊢ ((𝑏 = 𝑖 ∧ 𝑐 = ℎ) → (∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
20		eqid 2730	. . . 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 5500	. . 3 ⊢ (𝑖{⟨𝑏, 𝑐⟩ ∣ ∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑)}ℎ ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗))
22	7, 21	bitrdi 287	. 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 1218	. . . . 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 484	. . . . 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 2832	. . . 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 4887	. . . . . 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 1216	. . . . . 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 2765	. . . . 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 766	. . . . 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 2829	. . . 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 511	. . 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 1196	. . 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 3596	. 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 44281	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874   class class class wbr 5110  {copab 5172   × cxp 5639  Univcgru 10750   Coll ccoll 44246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-reg 9552  ax-inf2 9601  ax-ac2 10423

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-tc 9697  df-r1 9724  df-rank 9725  df-card 9899  df-cf 9901  df-acn 9902  df-ac 10076  df-wina 10644  df-ina 10645  df-gru 10751  df-scott 44232  df-coll 44247

This theorem is referenced by:  grumnueq  44283