Theorem en2lp 9521

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	⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)

Proof of Theorem en2lp

Step	Hyp	Ref	Expression
1		zfregfr 9519	. . 3 ⊢ E Fr V
2		efrn2lp 5604	. . 3 ⊢ (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpan 690	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
4		elex 3459	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5		elex 3459	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
6	4, 5	anim12i 613	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	6	con3i 154	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
8	3, 7	pm2.61i 182	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 395   ∈ wcel 2109  Vcvv 3438   E cep 5522   Fr wfr 5573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-reg 9503

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-eprel 5523  df-fr 5576

This theorem is referenced by:  elnanel  9522  cnvepnep  9523  elnotel  9525  preleqALT  9532  suc11reg  9534  axunndlem1  10508  axacndlem5  10524  bj-nsnid  37043  tratrb  44510  tratrbVD  44834