Theorem en3lp 8396

Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 38102 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 3878	. . . . 5 ⊢ ¬ 𝐶 ∈ ∅
2		eleq2 2677	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
3	1, 2	mtbiri 316	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4		tpid3g 4248	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	3, 4	nsyl 134	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ 𝐴)
6	5	intn3an3d 1436	. 2 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
7		tpex 6855	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} ∈ V
8		zfreg 8383	. . . 4 ⊢ (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
9	7, 8	mpan 702	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10		en3lplem2 8395	. . . . . 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 2798	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
13	12	rexlimiv 3009	. . 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 2865	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∩ cin 3539  ∅c0 3874  {ctp 4129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847  ax-reg 8380

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-nul 3875  df-sn 4126  df-pr 4128  df-tp 4130  df-uni 4373

This theorem is referenced by:  bj-inftyexpidisj  32274  tratrb  37767  tratrbVD  38119