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Theorem elpwincl1 30874
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 4542 . . 3 (𝐴 ∈ 𝒫 𝐶𝐴𝐶)
3 ssinss1 4171 . . 3 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
41, 2, 33syl 18 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
5 inex1g 5243 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
6 elpwg 4536 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
71, 5, 63syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
84, 7mpbird 256 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  Vcvv 3432  cin 3886  wss 3887  𝒫 cpw 4533
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535
This theorem is referenced by:  difelcarsg  32277  inelcarsg  32278  carsgclctunlem1  32284  carsgclctunlem2  32286  carsgclctunlem3  32287  carsgclctun  32288
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