Theorem snidb 3634

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 3633	. 2 |- ( A e. _V -> A e. { A } )
2		elex 2760	. 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 2158   _Vcvv 2749   {csn 3604

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sn 3610

This theorem is referenced by:  snid  3635