Theorem snidg 3647

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 2193	. 2 |- A = A
2		elsng 3633	. 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 1364   e. wcel 2164   {csn 3618

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sn 3624

This theorem is referenced by:  snidb  3648  elsn2g  3651  snnzg  3735  snmg  3736  exmidsssnc  4232  fvunsng  5752  fsnunfv  5759  1stconst  6274  2ndconst  6275  tfr0dm  6375  tfrlemibxssdm  6380  tfrlemi14d  6386  tfr1onlembxssdm  6396  tfr1onlemres  6402  tfrcllembxssdm  6409  tfrcllemres  6415  en1uniel  6858  onunsnss  6973  snon0  6994  supsnti  7064  fseq1p1m1  10160  elfzomin  10273  fsumsplitsnun  11562  divalgmod  12068  setsslid  12669  1strbas  12735  srnginvld  12767  lmodvscad  12785  mgm1  12953  mnd1id  13028  0subm  13056  cnpdis  14410  bj-sels  15406