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Theorem efrunt 30844
 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 5015 . . 3 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥 E 𝑥)
2 epel 4952 . . 3 (𝑥 E 𝑥𝑥𝑥)
31, 2sylnib 317 . 2 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑥)
43ralrimiva 2949 1 ( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   E cep 4947   Fr wfr 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-fr 4997 This theorem is referenced by: (None)
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