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Theorem bj-discrmoore 36296
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 5451 . . . . . 6 𝒫 𝐴 = 𝐴
21ineq1i 4209 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
3 inex1g 5320 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
4 inss1 4229 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
54a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
63, 5elpwd 4609 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
72, 6eqeltrid 2836 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
87adantr 480 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98bj-ismooredr 36294 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
10 pwexr 7755 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
119, 10impbii 208 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2105  Vcvv 3473  cin 3948  wss 3949  𝒫 cpw 4603   cuni 4909   cint 4951  Moorecmoore 36288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-int 4952  df-bj-moore 36289
This theorem is referenced by: (None)
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