Theorem en2lp 8510

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 8509	. . 3 ⊢ E Fr V
2		efrn2lp 5096	. . 3 ⊢ (( E Fr V ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
3	1, 2	mpan 706	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
4		elex 3212	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
5		elex 3212	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
6	4, 5	anim12i 590	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	6	con3i 150	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
8	3, 7	pm2.61i 176	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 384   ∈ wcel 1990  Vcvv 3200   E cep 5028   Fr wfr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073

This theorem is referenced by:  preleq  8514  suc11reg  8516  axunndlem1  9417  axacndlem5  9433  tratrb  38746  tratrbVD  39097