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Theorem bj-discrmoore 37094
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 5461 . . . . . 6 𝒫 𝐴 = 𝐴
21ineq1i 4224 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
3 inex1g 5325 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
4 inss1 4245 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
54a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
63, 5elpwd 4611 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
72, 6eqeltrid 2843 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
87adantr 480 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98bj-ismooredr 37092 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
10 pwexr 7784 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
119, 10impbii 209 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912   cint 4951  Moorecmoore 37086
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-int 4952  df-bj-moore 37087
This theorem is referenced by: (None)
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