Theorem elpwincl1 32812

Description: Closure of intersection 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
elpwincl1	⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶)

Proof of Theorem elpwincl1

Step	Hyp	Ref	Expression
1		elpwincl.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)
2		elpwi 4574	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶)
3		ssinss1 4206	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
4	1, 2, 3	3syl 19	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5		inex1g 5290	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V)
6		elpwg 4570	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → ((𝐴 ∩ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∩ 𝐵) ⊆ 𝐶))
7	1, 5, 6	3syl 19	. 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∩ 𝐵) ⊆ 𝐶))
8	4, 7	mpbird 260	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4569

This theorem is referenced by:  difelcarsg  34645  inelcarsg  34646  carsgclctunlem1  34652  carsgclctunlem2  34654  carsgclctunlem3  34655  carsgclctun  34656