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Theorem snnzg 4149
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 4058 . 2  |-  ( A  e.  V  ->  A  e.  { A } )
2 ne0i 3799 . 2  |-  ( A  e.  { A }  ->  { A }  =/=  (/) )
31, 2syl 16 1  |-  ( A  e.  V  ->  { A }  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    =/= wne 2652   (/)c0 3793   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-nul 3794  df-sn 4033
This theorem is referenced by:  snnz  4150  0nelop  4746  frirr  4865  frsn  5079  1stconst  6887  2ndconst  6888  fczsupp0  6947  hashge3el3dif  12528  pwsbas  14904  pwsle  14909  trnei  20519  uffix  20548  neiflim  20601  hausflim  20608  flimcf  20609  flimclslem  20611  cnpflf2  20627  cnpflf  20628  fclsfnflim  20654  ustneism  20852  ustuqtop5  20874  neipcfilu  20925  dv11cn  22528  usgra1v  24517  elpaddat  35671  elpadd2at  35673
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