Theorem snnzg 4779

Description: The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Assertion

Ref	Expression
snnzg	⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)

Proof of Theorem snnzg

Step	Hyp	Ref	Expression
1		snidg 4663	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2	1	ne0d 4336	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2107   ≠ wne 2941  ∅c0 4323  {csn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-dif 3952  df-nul 4324  df-sn 4630

This theorem is referenced by:  snn0d  4780  snnz  4781  frirr  5654  frsn  5764  omsucne  7874  1stconst  8086  2ndconst  8087  fczsupp0  8178  hashge3el3dif  14447  pwsbas  17433  pwsle  17438  trnei  23396  uffix  23425  neiflim  23478  flimclslem  23488  fclsfnflim  23531  ustneism  23728  ustuqtop5  23750  dv11cn  25518  noextendseq  27170  scutbdaylt  27319  lltropt  27367  snsssng  31752  cosnop  31917  elpadd2at  38677  onnog  42180  onnobdayg  42181  bdaybndbday  42183