Theorem snnz 3751

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 3749	. 2 |- ( A e. _V -> { A } =/= (/) )
3	1, 2	ax-mp 5	1 |- { A } =/= (/)

Syntax hints:   e. wcel 2175   =/= wne 2375   _Vcvv 2771   (/)c0 3459   {csn 3632

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-v 2773  df-dif 3167  df-nul 3460  df-sn 3638

This theorem is referenced by:  0nep0  4208  1n0  6508  ssfii  7058