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Theorem en2lp 9550
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 9549 . . 3 E Fr V
2 efrn2lp 5619 . . 3 (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴𝐵𝐵𝐴))
31, 2mpan 689 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
4 elex 3465 . . . 4 (𝐴𝐵𝐴 ∈ V)
5 elex 3465 . . . 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 3447   E cep 5540   Fr wfr 5589
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 5260  ax-nul 5267  ax-pr 5388  ax-reg 9536
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-eprel 5541  df-fr 5592
This theorem is referenced by:  elnanel  9551  cnvepnep  9552  elnotel  9554  preleqALT  9561  suc11reg  9563  axunndlem1  10539  axacndlem5  10555  bj-nsnid  35591  tratrb  42910  tratrbVD  43235
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