Theorem snid 3697

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Hypothesis

Ref	Expression
snid.1	|- A e. _V

Assertion

Ref	Expression
snid	|- A e. { A }

Proof of Theorem snid

Step	Hyp	Ref	Expression
1		snid.1	. 2 |- A e. _V
2		snidb 3696	. 2 |- ( A e. _V <-> A e. { A } )
3	1, 2	mpbi 145	1 |- A e. { A }

Syntax hints:   e. wcel 2200   _Vcvv 2799   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sn 3672

This theorem is referenced by:  vsnid  3698  exsnrex  3708  rabsnt  3741  sneqr  3838  undifexmid  4277  exmidexmid  4280  ss1o0el1  4281  exmidundif  4290  exmidundifim  4291  exmid1stab  4292  unipw  4303  intid  4310  ordtriexmidlem2  4612  ordtriexmid  4613  ontriexmidim  4614  ordtri2orexmid  4615  regexmidlem1  4625  0elsucexmid  4657  ordpwsucexmid  4662  opthprc  4770  fsn  5807  fsn2  5809  fvsn  5834  fvsnun1  5836  acexmidlema  5992  acexmidlemb  5993  acexmidlemab  5995  brtpos0  6398  mapsn  6837  mapsncnv  6842  0elixp  6876  en1  6951  djulclr  7216  djurclr  7217  djulcl  7218  djurcl  7219  djuf1olem  7220  exmidonfinlem  7371  elreal2  8017  1exp  10790  hashinfuni  10999  wrdexb  11083  0bits  12470  ennnfonelemhom  12986  dvef  15401  wlkl1loop  16069  djucllem  16164  bj-d0clsepcl  16288