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Theorem elpwdifcl 32545
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 4609 . . 3 (𝜑𝐴𝐶)
32ssdifssd 4147 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
4 difexg 5329 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
5 elpwg 4603 . . 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 2108  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-pw 4602
This theorem is referenced by:  pwldsys  34158  ldgenpisyslem1  34164  difelcarsg  34312  inelcarsg  34313  carsgclctunlem2  34321  carsgclctunlem3  34322  carsgclctun  34323
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