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Theorem elpwincl1 32553
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 4612 . . 3 (𝐴 ∈ 𝒫 𝐶𝐴𝐶)
3 ssinss1 4254 . . 3 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
41, 2, 33syl 18 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
5 inex1g 5325 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
6 elpwg 4608 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
71, 5, 63syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
84, 7mpbird 257 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607
This theorem is referenced by:  difelcarsg  34292  inelcarsg  34293  carsgclctunlem1  34299  carsgclctunlem2  34301  carsgclctunlem3  34302  carsgclctun  34303
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