Theorem grumnud 44713

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 2736	. 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 5680	. . . 4 ⊢ (𝑖(𝐺 × 𝐺)ℎ ↔ (𝑖 ∈ 𝐺 ∧ ℎ ∈ 𝐺))
5		brin 5137	. . . . 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 3433	. . . 4 ⊢ 𝑖 ∈ V
9		vex 3433	. . . 4 ⊢ ℎ ∈ V
10		simpr 484	. . . . . . . 8 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑑 = 𝑗)
11	10	unieqd 4863	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ∪ 𝑑 = ∪ 𝑗)
12		simplr 769	. . . . . . 7 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → 𝑐 = ℎ)
13	11, 12	eqeq12d 2752	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (∪ 𝑑 = 𝑐 ↔ ∪ 𝑗 = ℎ))
14		elequ1 2121	. . . . . . 7 ⊢ (𝑑 = 𝑗 → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
15	14	adantl 481	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑑 ∈ 𝑓 ↔ 𝑗 ∈ 𝑓))
16		eleq12 2826	. . . . . . 7 ⊢ ((𝑏 = 𝑖 ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
17	16	adantlr 716	. . . . . 6 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → (𝑏 ∈ 𝑑 ↔ 𝑖 ∈ 𝑗))
18	13, 15, 17	3anbi123d 1439	. . . . 5 ⊢ (((𝑏 = 𝑖 ∧ 𝑐 = ℎ) ∧ 𝑑 = 𝑗) → ((∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ (∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
19	18	cbvexdvaw 2041	. . . 4 ⊢ ((𝑏 = 𝑖 ∧ 𝑐 = ℎ) → (∃𝑑(∪ 𝑑 = 𝑐 ∧ 𝑑 ∈ 𝑓 ∧ 𝑏 ∈ 𝑑) ↔ ∃𝑗(∪ 𝑗 = ℎ ∧ 𝑗 ∈ 𝑓 ∧ 𝑖 ∈ 𝑗)))
20		eqid 2736	. . . 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 5492	. . 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 1219	. . . . 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 2839	. . . 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 4863	. . . . . 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 1217	. . . . . 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 2771	. . . . 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 767	. . . . 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 2836	. . . 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 1197	. . 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 3569	. 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 44712	1 ⊢ (𝜑 → 𝐺 ∈ 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850   class class class wbr 5085  {copab 5147   × cxp 5629  Univcgru 10713   Coll ccoll 44677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-tc 9656  df-r1 9688  df-rank 9689  df-card 9863  df-cf 9865  df-acn 9866  df-ac 10038  df-wina 10607  df-ina 10608  df-gru 10714  df-scott 44663  df-coll 44678

This theorem is referenced by:  grumnueq  44714