Theorem snid 3736

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 3735	. 2 |- ( A e. _V <-> A e. { A } )
3	1, 2	mpbi 145	1 |- A e. { A }

Syntax hints:   e. wcel 2209   _Vcvv 2821   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3711

This theorem is referenced by:  vsnid  3737  exsnrex  3747  rabsnt  3782  sneqr  3880  undifexmid  4325  exmidexmid  4328  ss1o0el1  4329  exmidundif  4338  exmidundifim  4339  exmid1stab  4340  unipw  4352  intid  4359  ordtriexmidlem2  4662  ordtriexmid  4663  ontriexmidim  4664  ordtri2orexmid  4665  regexmidlem1  4675  0elsucexmid  4707  ordpwsucexmid  4712  opthprc  4821  fsn  5871  fsn2  5873  fvsn  5901  fvsnun1  5903  acexmidlema  6066  acexmidlemb  6067  acexmidlemab  6069  brtpos0  6513  mapsn  6962  mapsncnv  6967  0elixp  7001  en1  7076  djulclr  7379  djurclr  7380  djulcl  7381  djurcl  7382  djuf1olem  7383  exmidonfinlem  7535  elreal2  8187  1exp  10983  hashinfuni  11194  wrdexb  11294  0bits  12704  ennnfonelemhom  13284  dvef  15751  wlkl1loop  16513  djucllem  16742  bj-d0clsepcl  16865