Theorem elpwdifcl 32344

Description: Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)

Hypothesis

Ref	Expression
elpwincl.1	⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)

Assertion

Ref	Expression
elpwdifcl	⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)

Proof of Theorem elpwdifcl

Step	Hyp	Ref	Expression
1		elpwincl.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)
2	1	elpwid 4615	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
3	2	ssdifssd 4143	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)
4		difexg 5333	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V)
5		elpwg 4609	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∈ V → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶))
6	1, 4, 5	3syl 18	. 2 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶))
7	3, 6	mpbird 256	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2098  Vcvv 3473   ∖ cdif 3946   ⊆ wss 3949  𝒫 cpw 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-pw 4608

This theorem is referenced by:  pwldsys  33809  ldgenpisyslem1  33815  difelcarsg  33963  inelcarsg  33964  carsgclctunlem2  33972  carsgclctunlem3  33973  carsgclctun  33974