Theorem mnugrud 40695

Description: Minimal universes are Grothendieck universes. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnugrud.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))))}
mnugrud.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)

Assertion

Ref	Expression
mnugrud	⊢ (𝜑 → 𝑈 ∈ Univ)

Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,𝑘,𝑚,𝑛,𝑝,𝑙

Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnugrud

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnugrud.1	. . 3 ⊢ 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		mnugrud.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3	1, 2	mnutrd 40691	. 2 ⊢ (𝜑 → Tr 𝑈)
4	2	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑈 ∈ 𝑀)
5		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
6	1, 4, 5	mnupwd 40678	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝒫 𝑥 ∈ 𝑈)
7	2	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → 𝑈 ∈ 𝑀)
8	5	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → 𝑥 ∈ 𝑈)
9		simpr 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈)
10	1, 7, 8, 9	mnuprd 40687	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ 𝑈) → {𝑥, 𝑦} ∈ 𝑈)
11	10	ralrimiva 3181	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
12	2	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (𝑈 ↑m 𝑥)) → 𝑈 ∈ 𝑀)
13	5	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (𝑈 ↑m 𝑥)) → 𝑥 ∈ 𝑈)
14		elmapi 8421	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑈 ↑m 𝑥) → 𝑦:𝑥⟶𝑈)
15	14	adantl 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (𝑈 ↑m 𝑥)) → 𝑦:𝑥⟶𝑈)
16	1, 12, 13, 15	mnurnd 40694	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (𝑈 ↑m 𝑥)) → ran 𝑦 ∈ 𝑈)
17	1, 12, 16	mnuunid 40688	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑈) ∧ 𝑦 ∈ (𝑈 ↑m 𝑥)) → ∪ ran 𝑦 ∈ 𝑈)
18	17	ralrimiva 3181	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)
19	6, 11, 18	3jca 1123	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
20	19	ralrimiva 3181	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))
21		elgrug 10207	. . 3 ⊢ (𝑈 ∈ 𝑀 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
22	2, 21	syl 17	. 2 ⊢ (𝜑 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))))
23	3, 20, 22	mpbir2and 711	1 ⊢ (𝜑 → 𝑈 ∈ Univ)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082  ∀wal 1534   = wceq 1536   ∈ wcel 2113  {cab 2798  ∀wral 3137  ∃wrex 3138   ⊆ wss 3929  𝒫 cpw 4532  {cpr 4562  ∪ cuni 4831  Tr wtr 5165  ran crn 5549  ⟶wf 6344  (class class class)co 7149   ↑_m cmap 8399  Univcgru 10205

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-reg 9049

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-fr 5507  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7682  df-2nd 7683  df-map 8401  df-gru 10206

This theorem is referenced by:  grumnueq  40698