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Theorem en3lp 8551
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 39394 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 3952 . . . . 5 ¬ 𝐶 ∈ ∅
2 eleq2 2719 . . . . 5 ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
31, 2mtbiri 316 . . . 4 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4 tpid3g 4337 . . . 4 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4nsyl 135 . . 3 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶𝐴)
65intn3an3d 1484 . 2 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
7 tpex 6999 . . . 4 {𝐴, 𝐵, 𝐶} ∈ V
8 zfreg 8541 . . . 4 (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
97, 8mpan 706 . . 3 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10 en3lplem2 8550 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1110com12 32 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1211necon2bd 2839 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)))
1312rexlimiv 3056 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
149, 13syl 17 . 2 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
156, 14pm2.61ine 2906 1 ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  cin 3606  c0 3948  {ctp 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215  df-uni 4469
This theorem is referenced by:  bj-inftyexpidisj  33227  tratrb  39063  tratrbVD  39411
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