Theorem snnz 3702

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 3700	. 2 |- ( A e. _V -> { A } =/= (/) )
3	1, 2	ax-mp 5	1 |- { A } =/= (/)

Syntax hints:   e. wcel 2141   =/= wne 2340   _Vcvv 2730   (/)c0 3414   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3589

This theorem is referenced by:  0nep0  4151  1n0  6411  ssfii  6951