Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ismooredr Structured version   Visualization version   GIF version

Theorem bj-ismooredr 34496
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 4531 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
2 bj-ismooredr.1 . . . . 5 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
32ex 416 . . . 4 (𝜑 → (𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
41, 3syl5 34 . . 3 (𝜑 → (𝑥 ∈ 𝒫 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
54ralrimiv 3176 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
6 bj-ismoore 34492 . 2 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
75, 6sylibr 237 1 (𝜑𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3133  cin 3918  wss 3919  𝒫 cpw 4522   cuni 4824   cint 4862  Moorecmoore 34490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-uni 4825  df-int 4863  df-bj-moore 34491
This theorem is referenced by:  bj-ismooredr2  34497  bj-discrmoore  34498
  Copyright terms: Public domain W3C validator