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Theorem bj-ismoored0 35987
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 35986 . 2 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
2 0elpw 5355 . . 3 ∅ ∈ 𝒫 𝐴
3 rint0 4995 . . . . 5 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
43eleq1d 2819 . . . 4 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
54rspcv 3609 . . 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 1542  wcel 2107  wral 3062  cin 3948  c0 4323  𝒫 cpw 4603   cuni 4909   cint 4951  Moorecmoore 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-uni 4910  df-int 4952  df-bj-moore 35985
This theorem is referenced by:  bj-0nmoore  35993
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