Theorem nelsn 4614

Description: If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)

Assertion

Ref	Expression
nelsn	⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Proof of Theorem nelsn

Step	Hyp	Ref	Expression
1		elsni 4588	. 2 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
2	1	necon3ai 2953	1 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐴 ∈ {𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   ≠ wne 2928  {csn 4571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-sn 4572

This theorem is referenced by:  frd  5568  fvn0fvelrn  6846  fvunsn  7108  fzdif1  13500  nnoddn2prmb  16720  chnccat  18527  lbsextlem4  21093  cnfldfun  21300  cnfldfunOLD  21313  obslbs  21662  logbgcd1irr  26726  upgrres1  29286  cycpmco2  33094  elrgspnlem4  33204  lindssn  33335  drngidlhash  33391  drnglidl1ne0  33432  drng0mxidl  33433  rsprprmprmidlb  33480  rprmirredb  33489  1arithufdlem4  33504  ig1pmindeg  33554  irngnminplynz  33717  algextdeglem4  33725  submateqlem1  33812  submateqlem2  33813  qqhval2  33987  derangsn  35206  ricdrng1  42561  prjspersym  42640  prjspreln0  42642  prjspnvs  42653  pr2eldif1  43587  pr2eldif2  43588  clsk3nimkb  44073  clsk1indlem1  44078  disjf1o  45228  cnrefiisplem  45867  fperdvper  45957  dvnmul  45981  wallispi  46108  etransc  46321  gsumge0cl  46409  meadjiunlem  46503  hspmbllem2  46665