Theorem snidg 3695

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 2229	. 2 |- A = A
2		elsng 3681	. 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 1395   e. wcel 2200   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sn 3672

This theorem is referenced by:  snidb  3696  elsn2g  3699  snnzg  3783  snmg  3784  exmidsssnc  4286  fvunsng  5832  fsnunfv  5839  1stconst  6365  2ndconst  6366  tfr0dm  6466  tfrlemibxssdm  6471  tfrlemi14d  6477  tfr1onlembxssdm  6487  tfr1onlemres  6493  tfrcllembxssdm  6500  tfrcllemres  6506  en1uniel  6954  onunsnss  7075  snon0  7098  supsnti  7168  fseq1p1m1  10286  elfzomin  10407  swrds1  11195  fsumsplitsnun  11925  divalgmod  12433  setsslid  13078  bassetsnn  13084  1strbas  13145  srnginvld  13178  lmodvscad  13196  mgm1  13398  mnd1id  13484  0subm  13512  cnpdis  14910  upgr1edc  15915  bj-sels  16235