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Theorem bj-ismoored0 37089
Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-ismoored0 (𝐴Moore 𝐴𝐴)

Proof of Theorem bj-ismoored0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bj-ismoore 37088 . 2 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
2 0elpw 5362 . . 3 ∅ ∈ 𝒫 𝐴
3 rint0 4993 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
43eleq1d 2824 . . . 4 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
54rspcv 3618 . . 3 (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
62, 5ax-mp 5 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴)
71, 6sylbi 217 1 (𝐴Moore 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  cin 3962  c0 4339  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
This theorem depends on definitions:  df-bi 207  df-an 396  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-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952  df-bj-moore 37087
This theorem is referenced by:  bj-0nmoore  37095
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