Theorem snidg 3636

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 2189	. 2 |- A = A
2		elsng 3622	. 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 2160   {csn 3607

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 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sn 3613

This theorem is referenced by:  snidb  3637  elsn2g  3640  snnzg  3724  snmg  3725  exmidsssnc  4221  fvunsng  5731  fsnunfv  5738  1stconst  6247  2ndconst  6248  tfr0dm  6348  tfrlemibxssdm  6353  tfrlemi14d  6359  tfr1onlembxssdm  6369  tfr1onlemres  6375  tfrcllembxssdm  6382  tfrcllemres  6388  en1uniel  6831  onunsnss  6946  snon0  6966  supsnti  7035  fseq1p1m1  10126  elfzomin  10238  fsumsplitsnun  11462  divalgmod  11967  setsslid  12566  1strbas  12632  srnginvld  12664  lmodvscad  12682  mgm1  12849  mnd1id  12923  0subm  12951  cnpdis  14219  bj-sels  15144