Step |
Hyp |
Ref |
Expression |
1 |
|
ordom |
⊢ Ord ω |
2 |
|
ordelss |
⊢ ( ( Ord ω ∧ 𝐴 ∈ ω ) → 𝐴 ⊆ ω ) |
3 |
1 2
|
mpan |
⊢ ( 𝐴 ∈ ω → 𝐴 ⊆ ω ) |
4 |
3
|
sspwd |
⊢ ( 𝐴 ∈ ω → 𝒫 𝐴 ⊆ 𝒫 ω ) |
5 |
4
|
sselda |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴 ) → 𝑎 ∈ 𝒫 ω ) |
6 |
|
nnfi |
⊢ ( 𝐴 ∈ ω → 𝐴 ∈ Fin ) |
7 |
|
elpwi |
⊢ ( 𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴 ) |
8 |
|
ssfi |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝑎 ⊆ 𝐴 ) → 𝑎 ∈ Fin ) |
9 |
6 7 8
|
syl2an |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴 ) → 𝑎 ∈ Fin ) |
10 |
5 9
|
elind |
⊢ ( ( 𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴 ) → 𝑎 ∈ ( 𝒫 ω ∩ Fin ) ) |
11 |
10
|
ex |
⊢ ( 𝐴 ∈ ω → ( 𝑎 ∈ 𝒫 𝐴 → 𝑎 ∈ ( 𝒫 ω ∩ Fin ) ) ) |
12 |
11
|
ssrdv |
⊢ ( 𝐴 ∈ ω → 𝒫 𝐴 ⊆ ( 𝒫 ω ∩ Fin ) ) |