Theorem snnzg 3811

Description: The singleton of a set is not empty. It is also inhabited as shown at snmg 3812. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
snnzg	|- ( A e. V -> { A } =/= (/) )

Proof of Theorem snnzg

Step	Hyp	Ref	Expression
1		snidg 3720	. 2 |- ( A e. V -> A e. { A } )
2		ne0i 3517	. 2 |- ( A e. { A } -> { A } =/= (/) )
3	1, 2	syl 14	1 |- ( A e. V -> { A } =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2205   =/= wne 2414   (/)c0 3510   {csn 3691

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-v 2817  df-dif 3215  df-nul 3511  df-sn 3697

This theorem is referenced by:  snnz  3813  0nelop  4366