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Theorem elpwdifcl 29246
 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 4148 . . 3 (𝜑𝐴𝐶)
32ssdifssd 3732 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
4 difexg 4778 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
5 elpwg 4144 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
61, 4, 53syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
73, 6mpbird 247 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1987  Vcvv 3190   ∖ cdif 3557   ⊆ wss 3560  𝒫 cpw 4136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574  df-pw 4138 This theorem is referenced by:  pwldsys  30043  ldgenpisyslem1  30049  difelcarsg  30195  inelcarsg  30196  carsgclctunlem2  30204  carsgclctunlem3  30205  carsgclctun  30206
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