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Theorem snid 3377
Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
Hypothesis
Ref Expression
snid.1 A V
Assertion
Ref Expression
snid A {A}

Proof of Theorem snid
StepHypRef Expression
1 snid.1 . 2 A V
2 snidb 3376 . 2 (A V ↔ A {A})
31, 2mpbi 133 1 A {A}
Colors of variables: wff set class
Syntax hints:   wcel 1374  Vcvv 2535  {csn 3350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sn 3356
This theorem is referenced by:  ssnid  3378  exsnrex  3387  rabsnt  3419  sneqr  3505  rext  3925  unipw  3927  intid  3934  snnex  4131  ordtriexmidlem2  4193  ordtriexmid  4194  ordtri2orexmid  4195  regexmidlem1  4202  ordpwsucexmid  4230  opthprc  4318  fsn  5260  fsn2  5262  fvsn  5283  fvsnun1  5285  acexmidlema  5427  acexmidlemb  5428  acexmidlemab  5430  brtpos0  5789  tfrlemi14  5869  elreal2  6542  bj-d0clsepcl  7148
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