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Theorem bj-ismoored0 36290
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 36289 . 2 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
2 0elpw 5353 . . 3 ∅ ∈ 𝒫 𝐴
3 rint0 4993 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
43eleq1d 2816 . . . 4 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
54rspcv 3607 . . 3 (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
62, 5ax-mp 5 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴)
71, 6sylbi 216 1 (𝐴Moore 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  wral 3059  cin 3946  c0 4321  𝒫 cpw 4601   cuni 4907   cint 4949  Moorecmoore 36287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-uni 4908  df-int 4950  df-bj-moore 36288
This theorem is referenced by:  bj-0nmoore  36296
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