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Theorem nelsn 4183
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 (𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})

Proof of Theorem nelsn
StepHypRef Expression
1 elsni 4165 . 2 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
21necon3ai 2815 1 (𝐴𝐵 → ¬ 𝐴 ∈ {𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 1987  wne 2790  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-sn 4149
This theorem is referenced by:  fvunsn  6399  nnoddn2prmb  15442  lbsextlem4  19080  cnfldfunALT  19678  obslbs  19993  upgrres1  26093  submateqlem1  29652  submateqlem2  29653  qqhval2  29805  derangsn  30857  clsk3nimkb  37817  clsk1indlem1  37822  disjf1o  38849  fperdvper  39436  dvnmul  39461  wallispi  39591  etransc  39804  gsumge0cl  39892  meadjiunlem  39986  hspmbllem2  40145
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