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Theorem bj-ismooredr 33949
Description: Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
Hypotheses
Ref Expression
bj-ismooredr.1 (𝜑𝐴𝑉)
bj-ismooredr.2 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
Assertion
Ref Expression
bj-ismooredr (𝜑𝐴Moore)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-ismooredr
StepHypRef Expression
1 elpwi 4426 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
2 bj-ismooredr.2 . . . . 5 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
32ex 405 . . . 4 (𝜑 → (𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
41, 3syl5 34 . . 3 (𝜑 → (𝑥 ∈ 𝒫 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
54ralrimiv 3124 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
6 bj-ismooredr.1 . . 3 (𝜑𝐴𝑉)
7 bj-ismoore 33944 . . 3 (𝐴𝑉 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
86, 7syl 17 . 2 (𝜑 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
95, 8mpbird 249 1 (𝜑𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wcel 2051  wral 3081  cin 3821  wss 3822  𝒫 cpw 4416   cuni 4708   cint 4745  Moorecmoore 33942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-in 3829  df-ss 3836  df-pw 4418  df-uni 4709  df-bj-moore 33943
This theorem is referenced by:  bj-discrmoore  33951
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