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Theorem snnzb 4611
 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 4610 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
2 df-ne 2952 . . . 4 ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
32con2bii 361 . . 3 ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
41, 3bitri 278 . 2 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
54con4bii 324 1 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409  ∅c0 4225  {csn 4522 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 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-dif 3861  df-nul 4226  df-sn 4523 This theorem is referenced by:  lpvtx  26960  loop1cycl  32615  elima4  33266
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