MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efrn2lp Structured version   Visualization version   GIF version

Theorem efrn2lp 5389
Description: A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)
Assertion
Ref Expression
efrn2lp (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))

Proof of Theorem efrn2lp
StepHypRef Expression
1 fr2nr 5385 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵 E 𝐶𝐶 E 𝐵))
2 epelg 5318 . . . 4 (𝐶𝐴 → (𝐵 E 𝐶𝐵𝐶))
3 epelg 5318 . . . 4 (𝐵𝐴 → (𝐶 E 𝐵𝐶𝐵))
42, 3bi2anan9r 627 . . 3 ((𝐵𝐴𝐶𝐴) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
54adantl 474 . 2 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 E 𝐶𝐶 E 𝐵) ↔ (𝐵𝐶𝐶𝐵)))
61, 5mtbid 316 1 (( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ¬ (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wcel 2050   class class class wbr 4929   E cep 5316   Fr wfr 5363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-eprel 5317  df-fr 5366
This theorem is referenced by:  en2lp  8864
  Copyright terms: Public domain W3C validator