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Theorem bj-discrmoore 34397
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 5335 . . . . . 6 𝒫 𝐴 = 𝐴
21ineq1i 4185 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
3 inex1g 5216 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
4 inss1 4205 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
54a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
63, 5elpwd 4550 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
72, 6eqeltrid 2917 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
87adantr 483 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98bj-ismooredr 34395 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
10 pwexr 7481 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
119, 10impbii 211 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wcel 2110  Vcvv 3495  cin 3935  wss 3936  𝒫 cpw 4539   cuni 4832   cint 4869  Moorecmoore 34389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-pw 4541  df-sn 4562  df-pr 4564  df-uni 4833  df-int 4870  df-bj-moore 34390
This theorem is referenced by: (None)
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