MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  epne3 Structured version   Visualization version   GIF version

Theorem epne3 7473
Description: A well-founded class contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
epne3 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))

Proof of Theorem epne3
StepHypRef Expression
1 fr3nr 7472 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵))
2 epelg 5442 . . . . 5 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
323ad2ant2 1130 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐵 E 𝐶𝐵𝐶))
4 epelg 5442 . . . . 5 (𝐷𝐴 → (𝐶 E 𝐷𝐶𝐷))
543ad2ant3 1131 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐶 E 𝐷𝐶𝐷))
6 epelg 5442 . . . . 5 (𝐵𝐴 → (𝐷 E 𝐵𝐷𝐵))
763ad2ant1 1129 . . . 4 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝐷 E 𝐵𝐷𝐵))
83, 5, 73anbi123d 1432 . . 3 ((𝐵𝐴𝐶𝐴𝐷𝐴) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
98adantl 484 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐷𝐷 E 𝐵) ↔ (𝐵𝐶𝐶𝐷𝐷𝐵)))
101, 9mtbid 326 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083  wcel 2114   class class class wbr 5042   E cep 5440   Fr wfr 5487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-eprel 5441  df-fr 5490
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator