Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-discrmoore Structured version   Visualization version   GIF version

Theorem bj-discrmoore 35282
Description: The powerclass 𝒫 𝐴 is a Moore collection if and only if 𝐴 is a set. It is then called the discrete Moore collection. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-discrmoore (𝐴 ∈ V ↔ 𝒫 𝐴Moore)

Proof of Theorem bj-discrmoore
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unipw 5366 . . . . . 6 𝒫 𝐴 = 𝐴
21ineq1i 4142 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
3 inex1g 5243 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
4 inss1 4162 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
54a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
63, 5elpwd 4541 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
72, 6eqeltrid 2843 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
87adantr 481 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98bj-ismooredr 35280 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
10 pwexr 7615 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
119, 10impbii 208 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2106  Vcvv 3432  cin 3886  wss 3887  𝒫 cpw 4533   cuni 4839   cint 4879  Moorecmoore 35274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840  df-int 4880  df-bj-moore 35275
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator