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Theorem en3lp 9558
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 43219 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 4294 . . . . 5 ¬ 𝐶 ∈ ∅
2 eleq2 2823 . . . . 5 ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
31, 2mtbiri 327 . . . 4 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4 tpid3g 4737 . . . 4 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4nsyl 140 . . 3 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶𝐴)
65intn3an3d 1482 . 2 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
7 tpex 7685 . . . 4 {𝐴, 𝐵, 𝐶} ∈ V
8 zfreg 9539 . . . 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 9557 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1110com12 32 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1211necon2bd 2956 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)))
1312rexlimiv 3142 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
149, 13syl 17 . 2 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
156, 14pm2.61ine 3025 1 ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wrex 3070  Vcvv 3447  cin 3913  c0 4286  {ctp 4594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-sn 4591  df-pr 4593  df-tp 4595  df-uni 4870
This theorem is referenced by:  bj-inftyexpidisj  35731  tratrb  42910  tratrbVD  43235
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