Theorem snidg 3423

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

Step	Hyp	Ref	Expression
1		eqid 2081	. 2 |- A = A
2		elsng 3413	. 2 |- ( A e. V -> ( A e. { A } <-> A = A ) )
3	1, 2	mpbiri 166	1 |- ( A e. V -> A e. { A } )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   {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:  snidb  3424  elsn2g  3427  snnzg  3507  snmg  3508  fvunsng  5378  fsnunfv  5384  1stconst  5862  2ndconst  5863  tfr0  5960  tfrlemibxssdm  5964  tfrlemi14d  5970  en1uniel  6307  onunsnss  6383  snon0  6387  supsnti  6418  fseq1p1m1  9111  elfzomin  9215  divalgmod  10327  bj-sels  10705