Theorem snidb 3606

Description: A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
snidb	|- ( A e. _V <-> A e. { A } )

Proof of Theorem snidb

Step	Hyp	Ref	Expression
1		snidg 3605	. 2 |- ( A e. _V -> A e. { A } )
2		elex 2737	. 2 |- ( A e. { A } -> A e. _V )
3	1, 2	impbii 125	1 |- ( A e. _V <-> A e. { A } )

Syntax hints:   <-> wb 104   e. wcel 2136   _Vcvv 2726   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sn 3582

This theorem is referenced by:  snid  3607