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Theorem bj-snmooreb 37115
Description: A singleton is a Moore collection, biconditional version. (Contributed by BJ, 9-Dec-2021.) (Proof shortened by BJ, 10-Apr-2024.)
Assertion
Ref Expression
bj-snmooreb (𝐴 ∈ V ↔ {𝐴} ∈ Moore)

Proof of Theorem bj-snmooreb
StepHypRef Expression
1 bj-snmoore 37114 . 2 (𝐴 ∈ V → {𝐴} ∈ Moore)
2 snprc 4717 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 216 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
4 bj-0nmoore 37113 . . . . 5 ¬ ∅ ∈ Moore
54a1i 11 . . . 4 𝐴 ∈ V → ¬ ∅ ∈ Moore)
63, 5eqneltrd 2861 . . 3 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
76con4i 114 . 2 ({𝐴} ∈ Moore𝐴 ∈ V)
81, 7impbii 209 1 (𝐴 ∈ V ↔ {𝐴} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  Moorecmoore 37104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908  df-int 4947  df-bj-moore 37105
This theorem is referenced by: (None)
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