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Theorem snidg 3431
Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
snidg  |-  ( A  e.  V  ->  A  e.  { A } )

Proof of Theorem snidg
StepHypRef Expression
1 eqid 2082 . 2  |-  A  =  A
2 elsng 3421 . 2  |-  ( A  e.  V  ->  ( A  e.  { A } 
<->  A  =  A ) )
31, 2mpbiri 166 1  |-  ( A  e.  V  ->  A  e.  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {csn 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sn 3412
This theorem is referenced by:  snidb  3432  elsn2g  3435  snnzg  3515  snmg  3516  fvunsng  5389  fsnunfv  5395  1stconst  5873  2ndconst  5874  tfr0dm  5971  tfrlemibxssdm  5976  tfrlemi14d  5982  tfr1onlembxssdm  5992  tfr1onlemres  5998  tfrcllembxssdm  6005  tfrcllemres  6011  en1uniel  6351  onunsnss  6437  snon0  6445  supsnti  6477  fseq1p1m1  9187  elfzomin  9292  divalgmod  10471  bj-sels  10863
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