Theorem snnzg 3507

Description: The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
snnzg	|- ( A e. V -> { A } =/= (/) )

Proof of Theorem snnzg

Step	Hyp	Ref	Expression
1		snidg 3423	. 2 |- ( A e. V -> A e. { A } )
2		ne0i 3257	. 2 |- ( A e. { A } -> { A } =/= (/) )
3	1, 2	syl 14	1 |- ( A e. V -> { A } =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1433   =/= wne 2245   (/)c0 3251   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-dif 2975  df-nul 3252  df-sn 3404

This theorem is referenced by:  snnz  3509  0nelop  4003