Theorem elpwdifcl 30816

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 4546	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
3	2	ssdifssd 4078	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)
4		difexg 5251	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V)
5		elpwg 4538	. . 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 2107  Vcvv 3427   ∖ cdif 3885   ⊆ wss 3888  𝒫 cpw 4535

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 2109  ax-9 2117  ax-ext 2708  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3071  df-v 3429  df-dif 3891  df-in 3895  df-ss 3905  df-pw 4537

This theorem is referenced by:  pwldsys  32067  ldgenpisyslem1  32073  difelcarsg  32219  inelcarsg  32220  carsgclctunlem2  32228  carsgclctunlem3  32229  carsgclctun  32230