Theorem snnz 3785

Description: The singleton of a set is not empty. It is also inhabited as shown at snm 3786. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
snnz.1	|- A e. _V

Assertion

Ref	Expression
snnz	|- { A } =/= (/)

Proof of Theorem snnz

Step	Hyp	Ref	Expression
1		snnz.1	. 2 |- A e. _V
2		snnzg 3783	. 2 |- ( A e. _V -> { A } =/= (/) )
3	1, 2	ax-mp 5	1 |- { A } =/= (/)

Syntax hints:   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   {csn 3666

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492  df-sn 3672

This theorem is referenced by:  0nep0  4248  1n0  6576  ssfii  7137