Theorem snidb 3624

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 3623	. 2 |- ( A e. _V -> A e. { A } )
2		elex 2750	. 2 |- ( A e. { A } -> A e. _V )
3	1, 2	impbii 126	1 |- ( A e. _V <-> A e. { A } )

Syntax hints:   <-> wb 105   e. wcel 2148   _Vcvv 2739   {csn 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sn 3600

This theorem is referenced by:  snid  3625