Theorem difelpw 5320

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 4075	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		elpw2g 5300	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐴 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐴))
3	1, 2	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048  𝒫 cpw 4560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-pw 4562

This theorem is used by:  satfvsuclem2  36326  clsk3nimkb  45295