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Theorem snnzb 4409
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 4408 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
2 df-ne 2938 . . . 4 ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
32con2bii 348 . . 3 ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
41, 3bitri 266 . 2 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
54con4bii 312 1 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197   = wceq 1652  wcel 2155  wne 2937  Vcvv 3350  c0 4079  {csn 4334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-nul 4080  df-sn 4335
This theorem is referenced by:  lpvtx  26240  elima4  32122
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