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Theorem elpwdifcl 29696
 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 4310 . . 3 (𝜑𝐴𝐶)
32ssdifssd 3899 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
4 difexg 4943 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
5 elpwg 4306 . . 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 2145  Vcvv 3351   ∖ cdif 3720   ⊆ wss 3723  𝒫 cpw 4298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-pw 4300 This theorem is referenced by:  pwldsys  30560  ldgenpisyslem1  30566  difelcarsg  30712  inelcarsg  30713  carsgclctunlem2  30721  carsgclctunlem3  30722  carsgclctun  30723
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