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Theorem elpwdifcl 32554
Description: Closure of class difference 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
elpwdifcl (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)

Proof of Theorem elpwdifcl
StepHypRef Expression
1 elpwincl.1 . . . 4 (𝜑𝐴 ∈ 𝒫 𝐶)
21elpwid 4614 . . 3 (𝜑𝐴𝐶)
32ssdifssd 4157 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
4 difexg 5335 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
5 elpwg 4608 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
61, 4, 53syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
73, 6mpbird 257 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  Vcvv 3478  cdif 3960  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-3an 1088  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-dif 3966  df-in 3970  df-ss 3980  df-pw 4607
This theorem is referenced by:  pwldsys  34138  ldgenpisyslem1  34144  difelcarsg  34292  inelcarsg  34293  carsgclctunlem2  34301  carsgclctunlem3  34302  carsgclctun  34303
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