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Theorem en2lp 9631
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 9630 . . 3 E Fr V
2 efrn2lp 5660 . . 3 (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴𝐵𝐵𝐴))
31, 2mpan 688 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
4 elex 3480 . . . 4 (𝐴𝐵𝐴 ∈ V)
5 elex 3480 . . . 4 (𝐵𝐴𝐵 ∈ V)
64, 5anim12i 611 . . 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 394  wcel 2098  Vcvv 3461   E cep 5581   Fr wfr 5630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-reg 9617
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-eprel 5582  df-fr 5633
This theorem is referenced by:  elnanel  9632  cnvepnep  9633  elnotel  9635  preleqALT  9642  suc11reg  9644  axunndlem1  10620  axacndlem5  10636  bj-nsnid  36680  tratrb  44117  tratrbVD  44442
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