Theorem snidg 3661

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 2204	. 2 |- A = A
2		elsng 3647	. 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 1372   e. wcel 2175   {csn 3632

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sn 3638

This theorem is referenced by:  snidb  3662  elsn2g  3665  snnzg  3749  snmg  3750  exmidsssnc  4246  fvunsng  5777  fsnunfv  5784  1stconst  6306  2ndconst  6307  tfr0dm  6407  tfrlemibxssdm  6412  tfrlemi14d  6418  tfr1onlembxssdm  6428  tfr1onlemres  6434  tfrcllembxssdm  6441  tfrcllemres  6447  en1uniel  6895  onunsnss  7013  snon0  7036  supsnti  7106  fseq1p1m1  10215  elfzomin  10333  fsumsplitsnun  11701  divalgmod  12209  setsslid  12854  1strbas  12920  srnginvld  12953  lmodvscad  12971  mgm1  13173  mnd1id  13259  0subm  13287  cnpdis  14685  bj-sels  15812