Step |
Hyp |
Ref |
Expression |
1 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∩ On ) ↔ ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On ) ) |
2 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) |
3 |
2
|
anbi2ci |
⊢ ( ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On ) ↔ ( 𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴 ) ) |
4 |
1 3
|
bitri |
⊢ ( 𝑥 ∈ ( 𝒫 𝐴 ∩ On ) ↔ ( 𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴 ) ) |
5 |
|
ordsssuc |
⊢ ( ( 𝑥 ∈ On ∧ Ord 𝐴 ) → ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴 ) ) |
6 |
5
|
expcom |
⊢ ( Ord 𝐴 → ( 𝑥 ∈ On → ( 𝑥 ⊆ 𝐴 ↔ 𝑥 ∈ suc 𝐴 ) ) ) |
7 |
6
|
pm5.32d |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴 ) ↔ ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) ) ) |
8 |
|
simpr |
⊢ ( ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) → 𝑥 ∈ suc 𝐴 ) |
9 |
|
ordsuc |
⊢ ( Ord 𝐴 ↔ Ord suc 𝐴 ) |
10 |
|
ordelon |
⊢ ( ( Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴 ) → 𝑥 ∈ On ) |
11 |
10
|
ex |
⊢ ( Ord suc 𝐴 → ( 𝑥 ∈ suc 𝐴 → 𝑥 ∈ On ) ) |
12 |
9 11
|
sylbi |
⊢ ( Ord 𝐴 → ( 𝑥 ∈ suc 𝐴 → 𝑥 ∈ On ) ) |
13 |
12
|
ancrd |
⊢ ( Ord 𝐴 → ( 𝑥 ∈ suc 𝐴 → ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) ) ) |
14 |
8 13
|
impbid2 |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥 ∈ suc 𝐴 ) ↔ 𝑥 ∈ suc 𝐴 ) ) |
15 |
7 14
|
bitrd |
⊢ ( Ord 𝐴 → ( ( 𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴 ) ↔ 𝑥 ∈ suc 𝐴 ) ) |
16 |
4 15
|
syl5bb |
⊢ ( Ord 𝐴 → ( 𝑥 ∈ ( 𝒫 𝐴 ∩ On ) ↔ 𝑥 ∈ suc 𝐴 ) ) |
17 |
16
|
eqrdv |
⊢ ( Ord 𝐴 → ( 𝒫 𝐴 ∩ On ) = suc 𝐴 ) |