MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snnzg Structured version   Unicode version

Theorem snnzg 3913
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
StepHypRef Expression
1 snidg 3831 . 2  |-  ( A  e.  V  ->  A  e.  { A } )
2 ne0i 3626 . 2  |-  ( A  e.  { A }  ->  { A }  =/=  (/) )
31, 2syl 16 1  |-  ( A  e.  V  ->  { A }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2598   (/)c0 3620   {csn 3806
This theorem is referenced by:  snnz  3914  0nelop  4438  frirr  4551  frsn  4940  xpimasn  5308  1stconst  6427  2ndconst  6428  pwsbas  13701  pwsle  13706  trnei  17916  uffix  17945  neiflim  17998  hausflim  18005  flimcf  18006  flimclslem  18008  cnpflf2  18024  cnpflf  18025  fclsfnflim  18051  ustneism  18245  ustuqtop5  18267  neipcfilu  18318  dv11cn  19877  usgra1v  21401  elpaddat  30538  elpadd2at  30540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-nul 3621  df-sn 3812
  Copyright terms: Public domain W3C validator