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Theorem bj-ismoorec 33388
Description: Characterization of Moore collections. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-ismoorec (𝐴Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-ismoorec
StepHypRef Expression
1 elex 3363 . 2 (𝐴Moore𝐴 ∈ V)
2 bj-ismoore 33387 . 2 (𝐴 ∈ V → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
31, 2biadan2 802 1 (𝐴Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wcel 2145  wral 3061  Vcvv 3351  cin 3722  𝒫 cpw 4297   cuni 4574   cint 4611  Moorecmoore 33385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-bj-moore 33386
This theorem is referenced by:  bj-ismoored  33390
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