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Theorem elpwincl1 32272
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 4604 . . 3 (𝐴 ∈ 𝒫 𝐶𝐴𝐶)
3 ssinss1 4232 . . 3 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
41, 2, 33syl 18 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
5 inex1g 5312 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
6 elpwg 4600 . . 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 205  wcel 2098  Vcvv 3468  cin 3942  wss 3943  𝒫 cpw 4597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599
This theorem is referenced by:  difelcarsg  33839  inelcarsg  33840  carsgclctunlem1  33846  carsgclctunlem2  33848  carsgclctunlem3  33849  carsgclctun  33850
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