Theorem snnzg 4777

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 4661	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2	1	ne0d 4334	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2940  ∅c0 4321  {csn 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-dif 3950  df-nul 4322  df-sn 4628

This theorem is referenced by:  snn0d  4778  snnz  4779  frirr  5652  frsn  5761  omsucne  7870  1stconst  8082  2ndconst  8083  fczsupp0  8174  hashge3el3dif  14443  pwsbas  17429  pwsle  17434  trnei  23387  uffix  23416  neiflim  23469  flimclslem  23479  fclsfnflim  23522  ustneism  23719  ustuqtop5  23741  dv11cn  25509  noextendseq  27159  scutbdaylt  27308  lltropt  27356  snsssng  31739  cosnop  31904  elpadd2at  38665  onnog  42165  onnobdayg  42166  bdaybndbday  42168