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Theorem snnzb 4718
Description: A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
Assertion
Ref Expression
snnzb (𝐴 ∈ V ↔ {𝐴} ≠ ∅)

Proof of Theorem snnzb
StepHypRef Expression
1 snprc 4717 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
2 df-ne 2937 . . . 4 ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
32con2bii 357 . . 3 ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
41, 3bitri 275 . 2 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
54con4bii 321 1 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1534  wcel 2099  wne 2936  Vcvv 3470  c0 4318  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-nul 4319  df-sn 4625
This theorem is referenced by:  lpvtx  28874  loop1cycl  34741  elima4  35365
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