Step |
Hyp |
Ref |
Expression |
1 |
|
difss |
⊢ ( 𝐴 ∖ 𝑥 ) ⊆ 𝐴 |
2 |
|
elpw2g |
⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 ↔ ( 𝐴 ∖ 𝑥 ) ⊆ 𝐴 ) ) |
3 |
1 2
|
mpbiri |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 ) |
4 |
|
distop |
⊢ ( 𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top ) |
5 |
|
unipw |
⊢ ∪ 𝒫 𝐴 = 𝐴 |
6 |
5
|
eqcomi |
⊢ 𝐴 = ∪ 𝒫 𝐴 |
7 |
6
|
iscld |
⊢ ( 𝒫 𝐴 ∈ Top → ( 𝑥 ∈ ( Clsd ‘ 𝒫 𝐴 ) ↔ ( 𝑥 ⊆ 𝐴 ∧ ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 ) ) ) |
8 |
4 7
|
syl |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ ( Clsd ‘ 𝒫 𝐴 ) ↔ ( 𝑥 ⊆ 𝐴 ∧ ( 𝐴 ∖ 𝑥 ) ∈ 𝒫 𝐴 ) ) ) |
9 |
3 8
|
mpbiran2d |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ ( Clsd ‘ 𝒫 𝐴 ) ↔ 𝑥 ⊆ 𝐴 ) ) |
10 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) |
11 |
9 10
|
bitr4di |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ ( Clsd ‘ 𝒫 𝐴 ) ↔ 𝑥 ∈ 𝒫 𝐴 ) ) |
12 |
11
|
eqrdv |
⊢ ( 𝐴 ∈ 𝑉 → ( Clsd ‘ 𝒫 𝐴 ) = 𝒫 𝐴 ) |