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Theorem bj-discrmoore 34903
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 5309 . . . . . 6 𝒫 𝐴 = 𝐴
21ineq1i 4099 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
3 inex1g 5187 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
4 inss1 4119 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
54a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
63, 5elpwd 4496 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
72, 6eqeltrid 2837 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
87adantr 484 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98bj-ismooredr 34901 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
10 pwexr 7506 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
119, 10impbii 212 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  Vcvv 3398  cin 3842  wss 3843  𝒫 cpw 4488   cuni 4796   cint 4836  Moorecmoore 34895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-pw 4490  df-sn 4517  df-pr 4519  df-uni 4797  df-int 4837  df-bj-moore 34896
This theorem is referenced by: (None)
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