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Theorem bj-ismooredr 36511
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 4605 . . . 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 3140 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
6 bj-ismoore 36507 . 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 3056  cin 3943  wss 3944  𝒫 cpw 4598   cuni 4903   cint 4944  Moorecmoore 36505
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 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359
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 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-pw 4600  df-uni 4904  df-int 4945  df-bj-moore 36506
This theorem is referenced by:  bj-ismooredr2  36512  bj-discrmoore  36513
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