Theorem snidg 3662

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 2205	. 2 |- A = A
2		elsng 3648	. 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 1373   e. wcel 2176   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sn 3639

This theorem is referenced by:  snidb  3663  elsn2g  3666  snnzg  3750  snmg  3751  exmidsssnc  4247  fvunsng  5778  fsnunfv  5785  1stconst  6307  2ndconst  6308  tfr0dm  6408  tfrlemibxssdm  6413  tfrlemi14d  6419  tfr1onlembxssdm  6429  tfr1onlemres  6435  tfrcllembxssdm  6442  tfrcllemres  6448  en1uniel  6896  onunsnss  7014  snon0  7037  supsnti  7107  fseq1p1m1  10216  elfzomin  10335  swrds1  11121  fsumsplitsnun  11730  divalgmod  12238  setsslid  12883  1strbas  12949  srnginvld  12982  lmodvscad  13000  mgm1  13202  mnd1id  13288  0subm  13316  cnpdis  14714  bj-sels  15850