Theorem difelpw 5243

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

Assertion

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

Proof of Theorem difelpw

Step	Hyp	Ref	Expression
1		difss 4046	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		elpw2g 5237	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐴))
3	1, 2	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2110   ∖ cdif 3863   ⊆ wss 3866  𝒫 cpw 4513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-pw 4515

This theorem is referenced by:  satfvsuclem2  33035  clsk3nimkb  41327