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Theorem elneq 9214
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 9213 . 2 ¬ 𝐵𝐵
2 nelelne 3040 . 2 𝐵𝐵 → (𝐴𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2110  wne 2940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-reg 9208
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-sn 4542  df-pr 4544
This theorem is referenced by:  nelaneq  9215  preleqg  9230  dfac2b  9744
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