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Theorem bj-snmooreb 34524
 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 34523 . 2 (𝐴 ∈ V → {𝐴} ∈ Moore)
2 snprc 4616 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 219 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
4 bj-0nmoore 34522 . . . . 5 ¬ ∅ ∈ Moore
54a1i 11 . . . 4 𝐴 ∈ V → ¬ ∅ ∈ Moore)
63, 5eqneltrd 2912 . . 3 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
76con4i 114 . 2 ({𝐴} ∈ Moore𝐴 ∈ V)
81, 7impbii 212 1 (𝐴 ∈ V ↔ {𝐴} ∈ Moore)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ∅c0 4246  {csn 4528  Moorecmoore 34513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4804  df-int 4842  df-bj-moore 34514 This theorem is referenced by: (None)
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