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Theorem elneq 9064
 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 9063 . 2 ¬ 𝐵𝐵
2 nelelne 3085 . 2 𝐵𝐵 → (𝐴𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴𝐵𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   ≠ wne 2987 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-reg 9058 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-sn 4529  df-pr 4531 This theorem is referenced by:  nelaneq  9065  preleqg  9080  dfac2b  9559
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