Theorem snidg 3652

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 2196	. 2 |- A = A
2		elsng 3638	. 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 2167   {csn 3623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sn 3629

This theorem is referenced by:  snidb  3653  elsn2g  3656  snnzg  3740  snmg  3741  exmidsssnc  4237  fvunsng  5759  fsnunfv  5766  1stconst  6288  2ndconst  6289  tfr0dm  6389  tfrlemibxssdm  6394  tfrlemi14d  6400  tfr1onlembxssdm  6410  tfr1onlemres  6416  tfrcllembxssdm  6423  tfrcllemres  6429  en1uniel  6872  onunsnss  6987  snon0  7010  supsnti  7080  fseq1p1m1  10186  elfzomin  10299  fsumsplitsnun  11601  divalgmod  12109  setsslid  12754  1strbas  12820  srnginvld  12852  lmodvscad  12870  mgm1  13072  mnd1id  13158  0subm  13186  cnpdis  14562  bj-sels  15644