Theorem snnz 3708

Description: The singleton of a set is not empty. (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 3706	. 2 |- ( A e. _V -> { A } =/= (/) )
3	1, 2	ax-mp 5	1 |- { A } =/= (/)

Syntax hints:   e. wcel 2146   =/= wne 2345   _Vcvv 2735   (/)c0 3420   {csn 3589

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-v 2737  df-dif 3129  df-nul 3421  df-sn 3595

This theorem is referenced by:  0nep0  4160  1n0  6423  ssfii  6963