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Theorem snnzb 4080
Description: A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)
Assertion
Ref Expression
snnzb  |-  ( A  e.  _V  <->  { A }  =/=  (/) )

Proof of Theorem snnzb
StepHypRef Expression
1 snprc 4079 . . 3  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 df-ne 2651 . . . 4  |-  ( { A }  =/=  (/)  <->  -.  { A }  =  (/) )
32con2bii 330 . . 3  |-  ( { A }  =  (/)  <->  -.  { A }  =/=  (/) )
41, 3bitri 249 . 2  |-  ( -.  A  e.  _V  <->  -.  { A }  =/=  (/) )
54con4bii 295 1  |-  ( A  e.  _V  <->  { A }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-nul 3784  df-sn 4017
This theorem is referenced by:  elima4  29452
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