Theorem snidg 3622

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 2177	. 2 |- A = A
2		elsng 3608	. 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 1353   e. wcel 2148   {csn 3593

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-sn 3599

This theorem is referenced by:  snidb  3623  elsn2g  3626  snnzg  3710  snmg  3711  exmidsssnc  4204  fvunsng  5711  fsnunfv  5718  1stconst  6222  2ndconst  6223  tfr0dm  6323  tfrlemibxssdm  6328  tfrlemi14d  6334  tfr1onlembxssdm  6344  tfr1onlemres  6350  tfrcllembxssdm  6357  tfrcllemres  6363  en1uniel  6804  onunsnss  6916  snon0  6935  supsnti  7004  fseq1p1m1  10094  elfzomin  10206  fsumsplitsnun  11427  divalgmod  11932  setsslid  12513  1strbas  12576  srnginvld  12608  lmodvscad  12626  mgm1  12789  mnd1id  12848  0subm  12871  cnpdis  13745  bj-sels  14669