Theorem snidb 3550

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 3549	. 2 |- ( A e. _V -> A e. { A } )
2		elex 2692	. 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 1480   _Vcvv 2681   {csn 3522

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sn 3528

This theorem is referenced by:  snid  3551