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Theorem en2lp 8371
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 8370 . . 3 E Fr V
2 efrn2lp 5010 . . 3 (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴𝐵𝐵𝐴))
31, 2mpan 702 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
4 elex 3185 . . . 4 (𝐴𝐵𝐴 ∈ V)
5 elex 3185 . . . 4 (𝐵𝐴𝐵 ∈ V)
64, 5anim12i 588 . . 3 ((𝐴𝐵𝐵𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
76con3i 149 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴𝐵𝐵𝐴))
83, 7pm2.61i 175 1 ¬ (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wcel 1977  Vcvv 3173   E cep 4937   Fr wfr 4984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828  ax-reg 8358
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4579  df-opab 4639  df-eprel 4939  df-fr 4987
This theorem is referenced by:  preleq  8375  suc11reg  8377  axunndlem1  9274  axacndlem5  9290  tratrb  37561  tratrbVD  37913
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