Theorem snidg 3589

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 2157	. 2 |- A = A
2		elsng 3575	. 2 |- ( A e. V -> ( A e. { A } <-> A = A ) )
3	1, 2	mpbiri 167	1 |- ( A e. V -> A e. { A } )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   {csn 3560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sn 3566

This theorem is referenced by:  snidb  3590  elsn2g  3593  snnzg  3676  snmg  3677  exmidsssnc  4164  fvunsng  5660  fsnunfv  5667  1stconst  6165  2ndconst  6166  tfr0dm  6266  tfrlemibxssdm  6271  tfrlemi14d  6277  tfr1onlembxssdm  6287  tfr1onlemres  6293  tfrcllembxssdm  6300  tfrcllemres  6306  en1uniel  6746  onunsnss  6858  snon0  6877  supsnti  6945  fseq1p1m1  9989  elfzomin  10098  fsumsplitsnun  11309  divalgmod  11810  setsslid  12211  1strbas  12260  srnginvld  12287  lmodvscad  12298  cnpdis  12613  bj-sels  13460