Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  efrunt Structured version   Visualization version   GIF version

Theorem efrunt 34512
Description: If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
Assertion
Ref Expression
efrunt ( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem efrunt
StepHypRef Expression
1 frirr 5646 . . 3 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥 E 𝑥)
2 epel 5576 . . 3 (𝑥 E 𝑥𝑥𝑥)
31, 2sylnib 327 . 2 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑥)
43ralrimiva 3145 1 ( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2106  wral 3060   class class class wbr 5141   E cep 5572   Fr wfr 5621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-eprel 5573  df-fr 5624
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator