Theorem difelpw 5318

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 4083	. 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
2		elpw2g 5298	. 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 2145   ∖ cdif 3896   ⊆ wss 3899  𝒫 cpw 4557

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-in 3906  df-ss 3916  df-pw 4559

This theorem is used by:  satfvsuclem2  35939  clsk3nimkb  44880