Theorem snid 3425

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 e. _V

Assertion

Ref	Expression
snid	|- A e. { A }

Proof of Theorem snid

Step	Hyp	Ref	Expression
1		snid.1	. 2 |- A e. _V
2		snidb 3424	. 2 |- ( A e. _V <-> A e. { A } )
3	1, 2	mpbi 143	1 |- A e. { A }

Syntax hints:   e. wcel 1433   _Vcvv 2601   {csn 3398

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sn 3404

This theorem is referenced by:  vsnid  3426  exsnrex  3435  rabsnt  3467  sneqr  3552  unipw  3972  intid  3979  ordtriexmidlem2  4264  ordtriexmid  4265  ordtri2orexmid  4266  regexmidlem1  4276  0elsucexmid  4308  ordpwsucexmid  4313  opthprc  4409  fsn  5356  fsn2  5358  fvsn  5379  fvsnun1  5381  acexmidlema  5523  acexmidlemb  5524  acexmidlemab  5526  brtpos0  5890  en1  6302  elreal2  6999  1exp  9505  bj-d0clsepcl  10720