Theorem en3lp 9077

Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 41199 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 4296	. . . . 5 ⊢ ¬ 𝐶 ∈ ∅
2		eleq2 2901	. . . . 5 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
3	1, 2	mtbiri 329	. . . 4 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4		tpid3g 4708	. . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
5	3, 4	nsyl 142	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ 𝐴)
6	5	intn3an3d 1477	. 2 ⊢ ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))
7		tpex 7470	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} ∈ V
8		zfreg 9059	. . . 4 ⊢ (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
9	7, 8	mpan 688	. . 3 ⊢ ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10		en3lplem2 9076	. . . . . 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 3032	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
13	12	rexlimiv 3280	. . 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 3100	1 ⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  Vcvv 3494   ∩ cin 3935  ∅c0 4291  {ctp 4571

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461  ax-reg 9056

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4568  df-pr 4570  df-tp 4572  df-uni 4839

This theorem is referenced by:  bj-inftyexpidisj  34495  tratrb  40890  tratrbVD  41215