Theorem elpwincl1 33003

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 4564	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → 𝐴 ⊆ 𝐶)
3		ssinss1 4191	. . 3 ⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
4	1, 2, 3	3syl 19	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
5		inex1g 5282	. . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 → (𝐴 ∩ 𝐵) ∈ V)
6		elpwg 4560	. . 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3906  df-ss 3916  df-pw 4559

This theorem is used by:  difelcarsg  34824  inelcarsg  34825  carsgclctunlem1  34831  carsgclctunlem2  34833  carsgclctunlem3  34834  carsgclctun  34835