Metamath Proof Explorer


Theorem pwin

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 ( 𝐴𝐵 ) = ( 𝒫 𝐴 ∩ 𝒫 𝐵 )

Proof

Step Hyp Ref Expression
1 ssin ( ( 𝑥𝐴𝑥𝐵 ) ↔ 𝑥 ⊆ ( 𝐴𝐵 ) )
2 velpw ( 𝑥 ∈ 𝒫 𝐴𝑥𝐴 )
3 velpw ( 𝑥 ∈ 𝒫 𝐵𝑥𝐵 )
4 2 3 anbi12i ( ( 𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵 ) ↔ ( 𝑥𝐴𝑥𝐵 ) )
5 velpw ( 𝑥 ∈ 𝒫 ( 𝐴𝐵 ) ↔ 𝑥 ⊆ ( 𝐴𝐵 ) )
6 1 4 5 3bitr4i ( ( 𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵 ) ↔ 𝑥 ∈ 𝒫 ( 𝐴𝐵 ) )
7 6 ineqri ( 𝒫 𝐴 ∩ 𝒫 𝐵 ) = 𝒫 ( 𝐴𝐵 )
8 7 eqcomi 𝒫 ( 𝐴𝐵 ) = ( 𝒫 𝐴 ∩ 𝒫 𝐵 )