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Theorem en2lp 9601
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 9600 . . 3 E Fr V
2 efrn2lp 5659 . . 3 (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴𝐵𝐵𝐴))
31, 2mpan 689 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
4 elex 3493 . . . 4 (𝐴𝐵𝐴 ∈ V)
5 elex 3493 . . . 4 (𝐵𝐴𝐵 ∈ V)
64, 5anim12i 614 . . 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 397  wcel 2107  Vcvv 3475   E cep 5580   Fr wfr 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-fr 5632
This theorem is referenced by:  elnanel  9602  cnvepnep  9603  elnotel  9605  preleqALT  9612  suc11reg  9614  axunndlem1  10590  axacndlem5  10606  bj-nsnid  35951  tratrb  43297  tratrbVD  43622
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