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Theorem en3lp 8684
Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 39597 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 4062 . . . . 5 ¬ 𝐶 ∈ ∅
2 eleq2 2828 . . . . 5 ({𝐴, 𝐵, 𝐶} = ∅ → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝐶 ∈ ∅))
31, 2mtbiri 316 . . . 4 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶 ∈ {𝐴, 𝐵, 𝐶})
4 tpid3g 4449 . . . 4 (𝐶𝐴𝐶 ∈ {𝐴, 𝐵, 𝐶})
53, 4nsyl 135 . . 3 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ 𝐶𝐴)
65intn3an3d 1593 . 2 ({𝐴, 𝐵, 𝐶} = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
7 tpex 7123 . . . 4 {𝐴, 𝐵, 𝐶} ∈ V
8 zfreg 8667 . . . 4 (({𝐴, 𝐵, 𝐶} ∈ V ∧ {𝐴, 𝐵, 𝐶} ≠ ∅) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
97, 8mpan 708 . . 3 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅)
10 en3lplem2 8683 . . . . . 6 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1110com12 32 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
1211necon2bd 2948 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ((𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)))
1312rexlimiv 3165 . . 3 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝑥 ∩ {𝐴, 𝐵, 𝐶}) = ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
149, 13syl 17 . 2 ({𝐴, 𝐵, 𝐶} ≠ ∅ → ¬ (𝐴𝐵𝐵𝐶𝐶𝐴))
156, 14pm2.61ine 3015 1 ¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  Vcvv 3340  cin 3714  c0 4058  {ctp 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115  ax-reg 8664
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-nul 4059  df-sn 4322  df-pr 4324  df-tp 4326  df-uni 4589
This theorem is referenced by:  bj-inftyexpidisj  33426  tratrb  39266  tratrbVD  39614
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