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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 (𝜑𝐴 ∈ 𝒫 𝐶)
Assertion
Ref Expression
elpwincl1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)

Proof of Theorem elpwincl1
StepHypRef Expression
1 elpwincl.1 . . 3 (𝜑𝐴 ∈ 𝒫 𝐶)
2 elpwi 4117 . . 3 (𝐴 ∈ 𝒫 𝐶𝐴𝐶)
3 ssinss1 3803 . . 3 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
41, 2, 33syl 18 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
5 inex1g 4729 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
6 elpwg 4116 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
71, 5, 63syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
84, 7mpbird 246 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 Colors of variables: wff setvar class 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
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