Theorem en2lp 9558

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 9556	. . 3 ⊢ E Fr V
2		efrn2lp 5626	. . 3 ⊢ (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpan 700	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
4		elex 3474	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5		elex 3474	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
6	4, 5	anim12i 622	. . 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 183	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 399   ∈ wcel 2141  Vcvv 3453   E cep 5544   Fr wfr 5595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-reg 9537

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-eprel 5545  df-fr 5598

This theorem is referenced by:  elnanel  9559  cnvepnep  9560  elnotel  9562  preleqALT  9569  suc11reg  9571  axunndlem1  10550  axacndlem5  10566  bj-nsnid  37519  tratrb  45076  tratrbVD  45400