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Theorem en3lp 8457
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 38563 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 3895 . . . . 5 ¬ 𝐶 ∈ ∅
2 eleq2 2687 . . . . 5 ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
31, 2mtbiri 317 . . . 4 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4 tpid3g 4275 . . . 4 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4nsyl 135 . . 3 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶𝐴)
65intn3an3d 1441 . 2 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
7 tpex 6910 . . . 4 {𝐴, 𝐵, 𝐶} ∈ V
8 zfreg 8444 . . . 4 (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
97, 8mpan 705 . . 3 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10 en3lplem2 8456 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1110com12 32 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1211necon2bd 2806 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)))
1312rexlimiv 3020 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
149, 13syl 17 . 2 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
156, 14pm2.61ine 2873 1 ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  Vcvv 3186  cin 3554  c0 3891  {ctp 4152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902  ax-reg 8441
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-nul 3892  df-sn 4149  df-pr 4151  df-tp 4153  df-uni 4403
This theorem is referenced by:  bj-inftyexpidisj  32730  tratrb  38228  tratrbVD  38580
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