Theorem difelpw 5299

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 4088	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		elpw2g 5278	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐴))
3	1, 2	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2113   ∖ cdif 3898   ⊆ wss 3901  𝒫 cpw 4554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-in 3908  df-ss 3918  df-pw 4556

This theorem is referenced by:  satfvsuclem2  35554  clsk3nimkb  44281