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Theorem bj-ismoored0 34931
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 34930 . 2 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
2 0elpw 5232 . . 3 ∅ ∈ 𝒫 𝐴
3 rint0 4888 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
43eleq1d 2818 . . . 4 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
54rspcv 3524 . . 3 (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
62, 5ax-mp 5 . 2 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴)
71, 6sylbi 220 1 (𝐴Moore 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wral 3054  cin 3852  c0 4221  𝒫 cpw 4498   cuni 4806   cint 4846  Moorecmoore 34928
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 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rab 3063  df-v 3402  df-dif 3856  df-in 3860  df-ss 3870  df-nul 4222  df-pw 4500  df-uni 4807  df-int 4847  df-bj-moore 34929
This theorem is referenced by:  bj-0nmoore  34937
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