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Theorem bj-ismooredr 36583
Description: Sufficient condition to be a Moore collection. Note that there is no sethood hypothesis on 𝐴: it is a consequence of the only hypothesis. (Contributed by BJ, 9-Dec-2021.)
Hypothesis
Ref Expression
bj-ismooredr.1 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
Assertion
Ref Expression
bj-ismooredr (𝜑𝐴Moore)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴

Proof of Theorem bj-ismooredr
StepHypRef Expression
1 elpwi 4606 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
2 bj-ismooredr.1 . . . . 5 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
32ex 412 . . . 4 (𝜑 → (𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
41, 3syl5 34 . . 3 (𝜑 → (𝑥 ∈ 𝒫 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
54ralrimiv 3141 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
6 bj-ismoore 36579 . 2 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
75, 6sylibr 233 1 (𝜑𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wral 3057  cin 3944  wss 3945  𝒫 cpw 4599   cuni 4904   cint 4945  Moorecmoore 36577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-in 3952  df-ss 3962  df-nul 4320  df-pw 4601  df-uni 4905  df-int 4946  df-bj-moore 36578
This theorem is referenced by:  bj-ismooredr2  36584  bj-discrmoore  36585
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