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Theorem efrunt 31928
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 5227 . . 3 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥 E 𝑥)
2 epel 5166 . . 3 (𝑥 E 𝑥𝑥𝑥)
31, 2sylnib 317 . 2 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑥)
43ralrimiva 3115 1 ( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 2145  wral 3061   class class class wbr 4787   E cep 5162   Fr wfr 5206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-eprel 5163  df-fr 5209
This theorem is referenced by: (None)
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