Theorem en2lp 9496

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 9494	. . 3 ⊢ E Fr V
2		efrn2lp 5597	. . 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 3457	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5		elex 3457	. . . 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 2111  Vcvv 3436   E cep 5515   Fr wfr 5566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-eprel 5516  df-fr 5569

This theorem is referenced by:  elnanel  9497  cnvepnep  9498  elnotel  9500  preleqALT  9507  suc11reg  9509  axunndlem1  10483  axacndlem5  10499  bj-nsnid  37103  tratrb  44568  tratrbVD  44892