Theorem elpwdifcl 32872

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 4571	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
3	2	ssdifssd 4101	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)
4		difexg 5300	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V)
5		elpwg 4565	. . 3 ⊢ ((𝐴 ∖ 𝐵) ∈ V → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶))
6	1, 4, 5	3syl 19	. 2 ⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∖ 𝐵) ⊆ 𝐶))
7	3, 6	mpbird 260	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  𝒫 cpw 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-pw 4564

This theorem is referenced by:  pwldsys  34547  ldgenpisyslem1  34553  difelcarsg  34700  inelcarsg  34701  carsgclctunlem2  34709  carsgclctunlem3  34710  carsgclctun  34711