Description: The power class of the intersection of two classes is the intersection of their power classes. Exercise 4.12(j) of Mendelson p. 235. (Contributed by NM, 23-Nov-2003)
Ref | Expression | ||
---|---|---|---|
Assertion | pwin | ⊢ 𝒫 ( 𝐴 ∩ 𝐵 ) = ( 𝒫 𝐴 ∩ 𝒫 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssin | ⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ) | |
2 | velpw | ⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 ) | |
3 | velpw | ⊢ ( 𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵 ) | |
4 | 2 3 | anbi12i | ⊢ ( ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵 ) ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵 ) ) |
5 | velpw | ⊢ ( 𝑥 ∈ 𝒫 ( 𝐴 ∩ 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 ∩ 𝐵 ) ) | |
6 | 1 4 5 | 3bitr4i | ⊢ ( ( 𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ 𝒫 𝐵 ) ↔ 𝑥 ∈ 𝒫 ( 𝐴 ∩ 𝐵 ) ) |
7 | 6 | ineqri | ⊢ ( 𝒫 𝐴 ∩ 𝒫 𝐵 ) = 𝒫 ( 𝐴 ∩ 𝐵 ) |
8 | 7 | eqcomi | ⊢ 𝒫 ( 𝐴 ∩ 𝐵 ) = ( 𝒫 𝐴 ∩ 𝒫 𝐵 ) |