Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpwdifcl Structured version   Visualization version   GIF version

Theorem elpwdifcl 29875
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 4362 . . 3 (𝜑𝐴𝐶)
32ssdifssd 3947 . 2 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
4 difexg 5004 . . 3 (𝐴 ∈ 𝒫 𝐶 → (𝐴𝐵) ∈ V)
5 elpwg 4358 . . 3 ((𝐴𝐵) ∈ V → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
61, 4, 53syl 18 . 2 (𝜑 → ((𝐴𝐵) ∈ 𝒫 𝐶 ↔ (𝐴𝐵) ⊆ 𝐶))
73, 6mpbird 249 1 (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  Vcvv 3386  cdif 3767  wss 3770  𝒫 cpw 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778  ax-sep 4976
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-v 3388  df-dif 3773  df-in 3777  df-ss 3784  df-pw 4352
This theorem is referenced by:  pwldsys  30735  ldgenpisyslem1  30741  difelcarsg  30887  inelcarsg  30888  carsgclctunlem2  30896  carsgclctunlem3  30897  carsgclctun  30898
  Copyright terms: Public domain W3C validator