Theorem snid 3722

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 3721	. 2 |- ( A e. _V <-> A e. { A } )
3	1, 2	mpbi 145	1 |- A e. { A }

Syntax hints:   e. wcel 2205   _Vcvv 2815   {csn 3691

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sn 3697

This theorem is referenced by:  vsnid  3723  exsnrex  3733  rabsnt  3768  sneqr  3866  undifexmid  4308  exmidexmid  4311  ss1o0el1  4312  exmidundif  4321  exmidundifim  4322  exmid1stab  4323  unipw  4335  intid  4342  ordtriexmidlem2  4644  ordtriexmid  4645  ontriexmidim  4646  ordtri2orexmid  4647  regexmidlem1  4657  0elsucexmid  4689  ordpwsucexmid  4694  opthprc  4803  fsn  5851  fsn2  5853  fvsn  5881  fvsnun1  5883  acexmidlema  6043  acexmidlemb  6044  acexmidlemab  6046  brtpos0  6485  mapsn  6927  mapsncnv  6932  0elixp  6966  en1  7041  djulclr  7342  djurclr  7343  djulcl  7344  djurcl  7345  djuf1olem  7346  exmidonfinlem  7498  elreal2  8147  1exp  10934  hashinfuni  11144  wrdexb  11240  0bits  12649  ennnfonelemhom  13183  dvef  15609  wlkl1loop  16370  djucllem  16589  bj-d0clsepcl  16712