Theorem en2lp 9620

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 9619	. . 3 ⊢ E Fr V
2		efrn2lp 5635	. . 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 3480	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5		elex 3480	. . . 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 2108  Vcvv 3459   E cep 5552   Fr wfr 5603

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-reg 9606

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-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-eprel 5553  df-fr 5606

This theorem is referenced by:  elnanel  9621  cnvepnep  9622  elnotel  9624  preleqALT  9631  suc11reg  9633  axunndlem1  10609  axacndlem5  10625  bj-nsnid  37088  tratrb  44561  tratrbVD  44885