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Theorem en2lp 9644
Description: No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
en2lp ¬ (𝐴𝐵𝐵𝐴)

Proof of Theorem en2lp
StepHypRef Expression
1 zfregfr 9643 . . 3 E Fr V
2 efrn2lp 5670 . . 3 (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴𝐵𝐵𝐴))
31, 2mpan 690 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
4 elex 3499 . . . 4 (𝐴𝐵𝐴 ∈ V)
5 elex 3499 . . . 4 (𝐵𝐴𝐵 ∈ V)
64, 5anim12i 613 . . 3 ((𝐴𝐵𝐵𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
76con3i 154 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
83, 7pm2.61i 182 1 ¬ (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wcel 2106  Vcvv 3478   E cep 5588   Fr wfr 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641
This theorem is referenced by:  elnanel  9645  cnvepnep  9646  elnotel  9648  preleqALT  9655  suc11reg  9657  axunndlem1  10633  axacndlem5  10649  bj-nsnid  37053  tratrb  44534  tratrbVD  44859
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