Step |
Hyp |
Ref |
Expression |
1 |
|
neiptop.o |
⊢ 𝐽 = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) } |
2 |
|
eleq1 |
⊢ ( 𝑎 = 𝐶 → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ) ) |
3 |
2
|
raleqbi1dv |
⊢ ( 𝑎 = 𝐶 → ( ∀ 𝑝 ∈ 𝑎 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ) ) |
4 |
3 1
|
elrab2 |
⊢ ( 𝐶 ∈ 𝐽 ↔ ( 𝐶 ∈ 𝒫 𝑋 ∧ ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ) ) |
5 |
|
0ex |
⊢ ∅ ∈ V |
6 |
|
eleq1 |
⊢ ( 𝐶 = ∅ → ( 𝐶 ∈ V ↔ ∅ ∈ V ) ) |
7 |
5 6
|
mpbiri |
⊢ ( 𝐶 = ∅ → 𝐶 ∈ V ) |
8 |
7
|
adantl |
⊢ ( ( ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ∧ 𝐶 = ∅ ) → 𝐶 ∈ V ) |
9 |
|
elex |
⊢ ( 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) → 𝐶 ∈ V ) |
10 |
9
|
ralimi |
⊢ ( ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) → ∀ 𝑝 ∈ 𝐶 𝐶 ∈ V ) |
11 |
|
r19.3rzv |
⊢ ( 𝐶 ≠ ∅ → ( 𝐶 ∈ V ↔ ∀ 𝑝 ∈ 𝐶 𝐶 ∈ V ) ) |
12 |
11
|
biimparc |
⊢ ( ( ∀ 𝑝 ∈ 𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅ ) → 𝐶 ∈ V ) |
13 |
10 12
|
sylan |
⊢ ( ( ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ∧ 𝐶 ≠ ∅ ) → 𝐶 ∈ V ) |
14 |
8 13
|
pm2.61dane |
⊢ ( ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) → 𝐶 ∈ V ) |
15 |
|
elpwg |
⊢ ( 𝐶 ∈ V → ( 𝐶 ∈ 𝒫 𝑋 ↔ 𝐶 ⊆ 𝑋 ) ) |
16 |
14 15
|
syl |
⊢ ( ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) → ( 𝐶 ∈ 𝒫 𝑋 ↔ 𝐶 ⊆ 𝑋 ) ) |
17 |
16
|
pm5.32ri |
⊢ ( ( 𝐶 ∈ 𝒫 𝑋 ∧ ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ) ↔ ( 𝐶 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ) ) |
18 |
4 17
|
bitri |
⊢ ( 𝐶 ∈ 𝐽 ↔ ( 𝐶 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ 𝐶 𝐶 ∈ ( 𝑁 ‘ 𝑝 ) ) ) |