Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-snmooreb Structured version   Visualization version   GIF version

Theorem bj-snmooreb 35516
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 35515 . 2 (𝐴 ∈ V → {𝐴} ∈ Moore)
2 snprc 4676 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 215 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
4 bj-0nmoore 35514 . . . . 5 ¬ ∅ ∈ Moore
54a1i 11 . . . 4 𝐴 ∈ V → ¬ ∅ ∈ Moore)
63, 5eqneltrd 2857 . . 3 𝐴 ∈ V → ¬ {𝐴} ∈ Moore)
76con4i 114 . 2 ({𝐴} ∈ Moore𝐴 ∈ V)
81, 7impbii 208 1 (𝐴 ∈ V ↔ {𝐴} ∈ Moore)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1541  wcel 2106  Vcvv 3443  c0 4280  {csn 4584  Moorecmoore 35505
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-pw 4560  df-sn 4585  df-pr 4587  df-uni 4864  df-int 4906  df-bj-moore 35506
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator