Theorem snid 3433

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 3432	. 2 |- ( A e. _V <-> A e. { A } )
3	1, 2	mpbi 143	1 |- A e. { A }

Syntax hints:   e. wcel 1434   _Vcvv 2602   {csn 3406

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sn 3412

This theorem is referenced by:  vsnid  3434  exsnrex  3443  rabsnt  3475  sneqr  3560  unipw  3980  intid  3987  ordtriexmidlem2  4272  ordtriexmid  4273  ordtri2orexmid  4274  regexmidlem1  4284  0elsucexmid  4316  ordpwsucexmid  4321  opthprc  4417  fsn  5367  fsn2  5369  fvsn  5390  fvsnun1  5392  acexmidlema  5534  acexmidlemb  5535  acexmidlemab  5537  brtpos0  5901  en1  6346  elreal2  7061  1exp  9602  sizeinfuni  9801  bj-d0clsepcl  10878