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Theorem elpwincl1 30200
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 4554 . . 3 (𝐴 ∈ 𝒫 𝐶𝐴𝐶)
3 ssinss1 4218 . . 3 (𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
41, 2, 33syl 18 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
5 inex1g 5220 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
6 elpwg 4548 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
71, 5, 63syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
84, 7mpbird 258 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2107  Vcvv 3500  cin 3939  wss 3940  𝒫 cpw 4542
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-in 3947  df-ss 3956  df-pw 4544
This theorem is referenced by:  difelcarsg  31454  inelcarsg  31455  carsgclctunlem1  31461  carsgclctunlem2  31463  carsgclctunlem3  31464  carsgclctun  31465
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