Theorem difelpw

Description: A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023)

	Ref	Expression
Assertion	difelpw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝐵 ) ∈ 𝒫 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		difss	⊢ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴
2		elpw2g	⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝐴 ∖ 𝐵 ) ∈ 𝒫 𝐴 ↔ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐴 ) )
3	1 2	mpbiri	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∖ 𝐵 ) ∈ 𝒫 𝐴 )