Metamath Proof Explorer


Theorem mnugrud

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

Ref Expression
Hypotheses mnugrud.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
mnugrud.2 ( 𝜑𝑈𝑀 )
Assertion mnugrud ( 𝜑𝑈 ∈ Univ )

Proof

Step Hyp Ref Expression
1 mnugrud.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
2 mnugrud.2 ( 𝜑𝑈𝑀 )
3 1 2 mnutrd ( 𝜑 → Tr 𝑈 )
4 2 adantr ( ( 𝜑𝑥𝑈 ) → 𝑈𝑀 )
5 simpr ( ( 𝜑𝑥𝑈 ) → 𝑥𝑈 )
6 1 4 5 mnupwd ( ( 𝜑𝑥𝑈 ) → 𝒫 𝑥𝑈 )
7 2 ad2antrr ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦𝑈 ) → 𝑈𝑀 )
8 5 adantr ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦𝑈 ) → 𝑥𝑈 )
9 simpr ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦𝑈 ) → 𝑦𝑈 )
10 1 7 8 9 mnuprd ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦𝑈 ) → { 𝑥 , 𝑦 } ∈ 𝑈 )
11 10 ralrimiva ( ( 𝜑𝑥𝑈 ) → ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 )
12 2 ad2antrr ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦 ∈ ( 𝑈m 𝑥 ) ) → 𝑈𝑀 )
13 5 adantr ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦 ∈ ( 𝑈m 𝑥 ) ) → 𝑥𝑈 )
14 elmapi ( 𝑦 ∈ ( 𝑈m 𝑥 ) → 𝑦 : 𝑥𝑈 )
15 14 adantl ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦 ∈ ( 𝑈m 𝑥 ) ) → 𝑦 : 𝑥𝑈 )
16 1 12 13 15 mnurnd ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦 ∈ ( 𝑈m 𝑥 ) ) → ran 𝑦𝑈 )
17 1 12 16 mnuunid ( ( ( 𝜑𝑥𝑈 ) ∧ 𝑦 ∈ ( 𝑈m 𝑥 ) ) → ran 𝑦𝑈 )
18 17 ralrimiva ( ( 𝜑𝑥𝑈 ) → ∀ 𝑦 ∈ ( 𝑈m 𝑥 ) ran 𝑦𝑈 )
19 6 11 18 3jca ( ( 𝜑𝑥𝑈 ) → ( 𝒫 𝑥𝑈 ∧ ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈m 𝑥 ) ran 𝑦𝑈 ) )
20 19 ralrimiva ( 𝜑 → ∀ 𝑥𝑈 ( 𝒫 𝑥𝑈 ∧ ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈m 𝑥 ) ran 𝑦𝑈 ) )
21 elgrug ( 𝑈𝑀 → ( 𝑈 ∈ Univ ↔ ( Tr 𝑈 ∧ ∀ 𝑥𝑈 ( 𝒫 𝑥𝑈 ∧ ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈m 𝑥 ) ran 𝑦𝑈 ) ) ) )
22 2 21 syl ( 𝜑 → ( 𝑈 ∈ Univ ↔ ( Tr 𝑈 ∧ ∀ 𝑥𝑈 ( 𝒫 𝑥𝑈 ∧ ∀ 𝑦𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈m 𝑥 ) ran 𝑦𝑈 ) ) ) )
23 3 20 22 mpbir2and ( 𝜑𝑈 ∈ Univ )