Theorem elpwdifcl 29358

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 4170	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
3	2	ssdifssd 3748	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐶)
4		difexg 4808	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∖ 𝐵) ∈ V)
5		elpwg 4166	. . 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 247	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  𝒫 cpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  pwldsys  30220  ldgenpisyslem1  30226  difelcarsg  30372  inelcarsg  30373  carsgclctunlem2  30381  carsgclctunlem3  30382  carsgclctun  30383