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Theorem snnzb 4652
 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 4651 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
2 df-ne 3021 . . . 4 ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
32con2bii 359 . . 3 ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
41, 3bitri 276 . 2 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
54con4bii 322 1 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1530   ∈ wcel 2106   ≠ wne 3020  Vcvv 3499  ∅c0 4294  {csn 4563 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-v 3501  df-dif 3942  df-nul 4295  df-sn 4564 This theorem is referenced by:  lpvtx  26768  loop1cycl  32269  elima4  32904
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