Theorem elpwincl1 31741

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 4608	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶)
3		ssinss1 4236	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
4	1, 2, 3	3syl 18	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5		inex1g 5318	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V)
6		elpwg 4604	. . 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 205   ∈ wcel 2107  Vcvv 3475   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3954  df-ss 3964  df-pw 4603

This theorem is referenced by:  difelcarsg  33247  inelcarsg  33248  carsgclctunlem1  33254  carsgclctunlem2  33256  carsgclctunlem3  33257  carsgclctun  33258