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Theorem elnel 8669
 Description: A class cannot be an element of one of its elements. (Contributed by AV, 14-Jun-2022.)
Assertion
Ref Expression
elnel (𝐴𝐵𝐵𝐴)

Proof of Theorem elnel
StepHypRef Expression
1 elnotel 8668 . 2 (𝐴𝐵 → ¬ 𝐵𝐴)
2 df-nel 3046 . 2 (𝐵𝐴 ↔ ¬ 𝐵𝐴)
31, 2sylibr 224 1 (𝐴𝐵𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2144   ∉ wnel 3045 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-reg 8652 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-eprel 5162  df-fr 5208 This theorem is referenced by: (None)
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