Step |
Hyp |
Ref |
Expression |
1 |
|
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 ) ) ) ) } |
2 |
|
mnugrud.2 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑀 ) |
3 |
1 2
|
mnutrd |
⊢ ( 𝜑 → Tr 𝑈 ) |
4 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝑈 ∈ 𝑀 ) |
5 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝑥 ∈ 𝑈 ) |
6 |
1 4 5
|
mnupwd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝒫 𝑥 ∈ 𝑈 ) |
7 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ 𝑈 ) → 𝑈 ∈ 𝑀 ) |
8 |
5
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ 𝑈 ) → 𝑥 ∈ 𝑈 ) |
9 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ 𝑈 ) → 𝑦 ∈ 𝑈 ) |
10 |
1 7 8 9
|
mnuprd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ 𝑈 ) → { 𝑥 , 𝑦 } ∈ 𝑈 ) |
11 |
10
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ) |
12 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ) → 𝑈 ∈ 𝑀 ) |
13 |
5
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ) → 𝑥 ∈ 𝑈 ) |
14 |
|
elmapi |
⊢ ( 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) → 𝑦 : 𝑥 ⟶ 𝑈 ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ) → 𝑦 : 𝑥 ⟶ 𝑈 ) |
16 |
1 12 13 15
|
mnurnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ) → ran 𝑦 ∈ 𝑈 ) |
17 |
1 12 16
|
mnuunid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) ∧ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ) → ∪ ran 𝑦 ∈ 𝑈 ) |
18 |
17
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → ∀ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ∪ ran 𝑦 ∈ 𝑈 ) |
19 |
6 11 18
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → ( 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ∪ ran 𝑦 ∈ 𝑈 ) ) |
20 |
19
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑈 ( 𝒫 𝑥 ∈ 𝑈 ∧ ∀ 𝑦 ∈ 𝑈 { 𝑥 , 𝑦 } ∈ 𝑈 ∧ ∀ 𝑦 ∈ ( 𝑈 ↑m 𝑥 ) ∪ ran 𝑦 ∈ 𝑈 ) ) |
21 |
|
elgrug |
⊢ ( 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 |
⊢ ( 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 |
⊢ ( 𝜑 → 𝑈 ∈ Univ ) |