Theorem grumnud 44860

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 2765	. 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 5701	. . . 4 ⊢ (𝑖(𝐺 × 𝐺)ℎ ↔ (𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺))
5		brin 5157	. . . . 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 547	. . . 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 3461	. . . 4 ⊢ 𝑖 ∈ V
9		vex 3461	. . . 4 ⊢ ℎ ∈ V
10		simpr 489	. . . . . . . 8 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
11	10	unieqd 4881	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ∪ 𝑑 = ∪ 𝑗)
12		simplr 780	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑐 = ℎ)
13	11, 12	eqeq12d 2781	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (∪ 𝑑 = 𝑐 ↔ ∪ 𝑗 = ℎ))
14		elequ1 2152	. . . . . . 7 ⊢ (𝑑 = 𝑗 → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
15	14	adantl 486	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
16		eleq12 2855	. . . . . . 7 ⊢ ((𝑏 = 𝑖 ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
17	16	adantlr 727	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
18	13, 15, 17	3anbi123d 1460	. . . . 5 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ((∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
19	18	cbvexdvaw 2062	. . . 4 ⊢ ((𝑏 = 𝑖 ∧ 𝑐 = ℎ) → (∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
20		eqid 2765	. . . 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 5512	. . 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 1234	. . . . 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 489	. . . . 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 2868	. . . 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 4881	. . . . . 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 1232	. . . . . 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 2800	. . . . 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 778	. . . . 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 2865	. . . 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 520	. . 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 1212	. . 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 3589	. 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 44859	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089   ∩ cin 3906   ⊆ wss 3907  𝒫 cpw 4558  ∪ cuni 4868   class class class wbr 5105  {copab 5167   × cxp 5650  Univcgru 10763   Coll ccoll 44824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-reg 9542  ax-inf2 9598  ax-ac2 10435

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-tc 9692  df-r1 9724  df-rank 9725  df-scott 9846  df-card 9913  df-cf 9915  df-acn 9916  df-ac 10088  df-wina 10657  df-ina 10658  df-gru 10764  df-coll 44825

This theorem is referenced by:  grumnueq  44861