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Theorem bj-discrmoore 33515
Description: The discrete Moore collection on a set. (Contributed by BJ, 9-Dec-2021.)
Assertion
Ref Expression
bj-discrmoore (𝐴 ∈ V ↔ 𝒫 𝐴Moore)

Proof of Theorem bj-discrmoore
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pwexg 5016 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 unipw 5076 . . . . . 6 𝒫 𝐴 = 𝐴
32ineq1i 3974 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
4 inex1g 4964 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
5 inss1 3994 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
65a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
74, 6elpwd 4326 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
83, 7syl5eqel 2848 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98adantr 472 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
101, 9bj-ismooredr 33513 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
11 pwexr 7176 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
1210, 11impbii 200 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2155  Vcvv 3350  cin 3733  wss 3734  𝒫 cpw 4317   cuni 4596   cint 4635  Moorecmoore 33506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-pw 4319  df-sn 4337  df-pr 4339  df-uni 4597  df-bj-moore 33507
This theorem is referenced by: (None)
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