Theorem en3lp 9511

Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 44961 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 4287	. . . . 5 ⊢ ¬ 𝐶 ∈ ∅
2		eleq2 2822	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
3	1, 2	mtbiri 327	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4		tpid3g 4724	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	3, 4	nsyl 140	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ 𝐴)
6	5	intn3an3d 1483	. 2 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
7		tpex 7685	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} ∈ V
8		zfreg 9489	. . . 4 ⊢ (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
9	7, 8	mpan 690	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10		en3lplem2 9510	. . . . . 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 2945	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
13	12	rexlimiv 3127	. . 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 3012	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∃wrex 3057  Vcvv 3437   ∩ cin 3897  ∅c0 4282  {ctp 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674  ax-reg 9485

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-sn 4576  df-pr 4578  df-tp 4580  df-uni 4859

This theorem is referenced by:  bj-inftyexpidisj  37275  tratrb  44653  tratrbVD  44977