Theorem snnzg 3723

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 3635	. 2 |- ( A e. V -> A e. { A } )
2		ne0i 3443	. 2 |- ( A e. { A } -> { A } =/= (/) )
3	1, 2	syl 14	1 |- ( A e. V -> { A } =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2159   =/= wne 2359   (/)c0 3436   {csn 3606

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-v 2753  df-dif 3145  df-nul 3437  df-sn 3612

This theorem is referenced by:  snnz  3725  0nelop  4262