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Theorem en3lp 9064
 Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 41594 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
en3lp ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

Proof of Theorem en3lp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 noel 4247 . . . . 5 ¬ 𝐶 ∈ ∅
2 eleq2 2878 . . . . 5 ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
31, 2mtbiri 330 . . . 4 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4 tpid3g 4668 . . . 4 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4nsyl 142 . . 3 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶𝐴)
65intn3an3d 1478 . 2 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
7 tpex 7453 . . . 4 {𝐴, 𝐵, 𝐶} ∈ V
8 zfreg 9046 . . . 4 (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
97, 8mpan 689 . . 3 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10 en3lplem2 9063 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1110com12 32 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1211necon2bd 3003 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)))
1312rexlimiv 3239 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
149, 13syl 17 . 2 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
156, 14pm2.61ine 3070 1 ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  Vcvv 3441   ∩ cin 3880  ∅c0 4243  {ctp 4529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444  ax-reg 9043 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-tp 4530  df-uni 4802 This theorem is referenced by:  bj-inftyexpidisj  34644  tratrb  41285  tratrbVD  41610
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