Theorem difelpw 5269

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 4062	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		elpw2g 5263	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐴))
3	1, 2	mpbiri 257	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2108   ∖ cdif 3880   ⊆ wss 3883  𝒫 cpw 4530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-pw 4532

This theorem is referenced by:  satfvsuclem2  33222  clsk3nimkb  41539