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Theorem bj-ismoored0 33393
Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-ismoored0 (𝐴Moore 𝐴𝐴)

Proof of Theorem bj-ismoored0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 bj-ismoore 33391 . . 3 (𝐴Moore → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
2 0elpw 4965 . . . 4 ∅ ∈ 𝒫 𝐴
3 rint0 4651 . . . . . 6 (𝑥 = ∅ → ( 𝐴 𝑥) = 𝐴)
43eleq1d 2835 . . . . 5 (𝑥 = ∅ → (( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
54rspcv 3456 . . . 4 (∅ ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴))
62, 5ax-mp 5 . . 3 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 𝐴𝐴)
71, 6syl6bi 243 . 2 (𝐴Moore → (𝐴Moore 𝐴𝐴))
87pm2.43i 52 1 (𝐴Moore 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061  cin 3722  c0 4063  𝒫 cpw 4297   cuni 4574   cint 4611  Moorecmoore 33389
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  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-uni 4575  df-int 4612  df-bj-moore 33390
This theorem is referenced by:  bj-0nmoore  33399
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