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Theorem elneq 9641
Description: A class is not equal to any of its elements. (Contributed by AV, 14-Jun-2022.)
Assertion
Ref Expression
elneq (𝐴𝐵𝐴𝐵)

Proof of Theorem elneq
StepHypRef Expression
1 elirr 9640 . 2 ¬ 𝐵𝐵
2 nelelne 3031 . 2 𝐵𝐵 → (𝐴𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2099  wne 2930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-pr 5433  ax-reg 9635
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-v 3464  df-un 3952  df-sn 4634  df-pr 4636
This theorem is referenced by:  nelaneq  9642  preleqg  9658  dfac2b  10173  disjressuc2  38086  oaomoencom  42983  oenassex  42984  tfsconcat0b  43012
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