Theorem elpwincl1 32506

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 4582	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶)
3		ssinss1 4221	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5		inex1g 5289	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V)
6		elpwg 4578	. . 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 257	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-pw 4577

This theorem is referenced by:  difelcarsg  34342  inelcarsg  34343  carsgclctunlem1  34349  carsgclctunlem2  34351  carsgclctunlem3  34352  carsgclctun  34353