Theorem en3lp 8551

Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 39394 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
en3lp	⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Proof of Theorem en3lp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3952	. . . . 5 ⊢ ¬ 𝐶 ∈ ∅
2		eleq2 2719	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
3	1, 2	mtbiri 316	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4		tpid3g 4337	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	3, 4	nsyl 135	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ 𝐴)
6	5	intn3an3d 1484	. 2 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
7		tpex 6999	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} ∈ V
8		zfreg 8541	. . . 4 ⊢ (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
9	7, 8	mpan 706	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10		en3lplem2 8550	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
11	10	com12 32	. . . . 5 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
12	11	necon2bd 2839	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
13	12	rexlimiv 3056	. . 3 ⊢ (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
14	9, 13	syl 17	. 2 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
15	6, 14	pm2.61ine 2906	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942  Vcvv 3231   ∩ cin 3606  ∅c0 3948  {ctp 4214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215  df-uni 4469

This theorem is referenced by:  bj-inftyexpidisj  33227  tratrb  39063  tratrbVD  39411