Theorem snidg 3698

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 2231	. 2 |- A = A
2		elsng 3684	. 2 |- ( A e. V -> ( A e. { A } <-> A = A ) )
3	1, 2	mpbiri 168	1 |- ( A e. V -> A e. { A } )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {csn 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sn 3675

This theorem is referenced by:  snidb  3699  elsn2g  3702  snnzg  3789  snmg  3790  exmidsssnc  4293  fvunsng  5847  fsnunfv  5854  1stconst  6385  2ndconst  6386  tfr0dm  6487  tfrlemibxssdm  6492  tfrlemi14d  6498  tfr1onlembxssdm  6508  tfr1onlemres  6514  tfrcllembxssdm  6521  tfrcllemres  6527  en1uniel  6977  onunsnss  7108  snon0  7133  supsnti  7203  fseq1p1m1  10328  elfzomin  10450  swrds1  11248  fsumsplitsnun  11979  divalgmod  12487  setsslid  13132  bassetsnn  13138  1strbas  13199  srnginvld  13232  lmodvscad  13250  mgm1  13452  mnd1id  13538  0subm  13566  cnpdis  14965  upgr1edc  15971  uspgr1edc  16090  vtxd0nedgbfi  16149  1loopgrvd2fi  16155  1hegrvtxdg1fi  16159  wlk1walkdom  16209  bj-sels  16509