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Theorem bj-discrmoore 33390
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 4999 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
2 unipw 5067 . . . . . 6 𝒫 𝐴 = 𝐴
32ineq1i 3953 . . . . 5 ( 𝒫 𝐴 𝑥) = (𝐴 𝑥)
4 inex1g 4953 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ∈ V)
5 inss1 3976 . . . . . . 7 (𝐴 𝑥) ⊆ 𝐴
65a1i 11 . . . . . 6 (𝐴 ∈ V → (𝐴 𝑥) ⊆ 𝐴)
74, 6elpwd 4311 . . . . 5 (𝐴 ∈ V → (𝐴 𝑥) ∈ 𝒫 𝐴)
83, 7syl5eqel 2843 . . . 4 (𝐴 ∈ V → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
98adantr 472 . . 3 ((𝐴 ∈ V ∧ 𝑥 ⊆ 𝒫 𝐴) → ( 𝒫 𝐴 𝑥) ∈ 𝒫 𝐴)
101, 9bj-ismooredr 33388 . 2 (𝐴 ∈ V → 𝒫 𝐴Moore)
11 pwexr 7140 . 2 (𝒫 𝐴Moore𝐴 ∈ V)
1210, 11impbii 199 1 (𝐴 ∈ V ↔ 𝒫 𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 2139  Vcvv 3340  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588   cint 4627  Moorecmoore 33381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-bj-moore 33382
This theorem is referenced by: (None)
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