Step |
Hyp |
Ref |
Expression |
1 |
|
mnupwd.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 |
|
mnupwd.2 |
⊢ ( 𝜑 → 𝑈 ∈ 𝑀 ) |
3 |
|
mnupwd.3 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
4 |
|
0ex |
⊢ ∅ ∈ V |
5 |
4
|
a1i |
⊢ ( 𝜑 → ∅ ∈ V ) |
6 |
1 2 3 5
|
mnuop23d |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅ ) → ∃ 𝑢 ∈ ∅ ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) |
7 |
|
simpl |
⊢ ( ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅ ) → ∃ 𝑢 ∈ ∅ ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) → 𝒫 𝐴 ⊆ 𝑤 ) |
8 |
7
|
reximi |
⊢ ( ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅ ) → ∃ 𝑢 ∈ ∅ ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) → ∃ 𝑤 ∈ 𝑈 𝒫 𝐴 ⊆ 𝑤 ) |
9 |
6 8
|
syl |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑈 𝒫 𝐴 ⊆ 𝑤 ) |
10 |
1 2 9
|
mnuss2d |
⊢ ( 𝜑 → 𝒫 𝐴 ∈ 𝑈 ) |