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Theorem snnzb 4684
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 4683 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
2 df-ne 2945 . . . 4 ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
32con2bii 358 . . 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 1542  wcel 2107  wne 2944  Vcvv 3448  c0 4287  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-nul 4288  df-sn 4592
This theorem is referenced by:  lpvtx  28061  loop1cycl  33771  elima4  34389
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