Theorem elpwdifcl 32462

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 4575	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
3	2	ssdifssd 4113	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)
4		difexg 5287	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V)
5		elpwg 4569	. . 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 257	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  𝒫 cpw 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-pw 4568

This theorem is referenced by:  pwldsys  34154  ldgenpisyslem1  34160  difelcarsg  34308  inelcarsg  34309  carsgclctunlem2  34317  carsgclctunlem3  34318  carsgclctun  34319