Theorem elpwincl1 28741

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 4117	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶)
3		ssinss1 3803	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5		inex1g 4729	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V)
6		elpwg 4116	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → ((𝐴 ∩ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∩ 𝐵) ⊆ 𝐶))
7	1, 5, 6	3syl 18	. 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∈ 𝒫 𝐶 ↔ (𝐴 ∩ 𝐵) ⊆ 𝐶))
8	4, 7	mpbird 246	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110

This theorem is referenced by:  difelcarsg  29699  inelcarsg  29700  carsgclctunlem1  29706  carsgclctunlem2  29708  carsgclctunlem3  29709  carsgclctun  29710