Theorem snidg 3672

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 2207	. 2 |- A = A
2		elsng 3658	. 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 2178   {csn 3643

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sn 3649

This theorem is referenced by:  snidb  3673  elsn2g  3676  snnzg  3760  snmg  3761  exmidsssnc  4263  fvunsng  5801  fsnunfv  5808  1stconst  6330  2ndconst  6331  tfr0dm  6431  tfrlemibxssdm  6436  tfrlemi14d  6442  tfr1onlembxssdm  6452  tfr1onlemres  6458  tfrcllembxssdm  6465  tfrcllemres  6471  en1uniel  6919  onunsnss  7040  snon0  7063  supsnti  7133  fseq1p1m1  10251  elfzomin  10372  swrds1  11159  fsumsplitsnun  11845  divalgmod  12353  setsslid  12998  1strbas  13064  srnginvld  13097  lmodvscad  13115  mgm1  13317  mnd1id  13403  0subm  13431  cnpdis  14829  upgr1edc  15829  bj-sels  16049