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Theorem efrunt 32836
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 5525 . . 3 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥 E 𝑥)
2 epel 5462 . . 3 (𝑥 E 𝑥𝑥𝑥)
31, 2sylnib 329 . 2 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑥)
43ralrimiva 3179 1 ( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2105  wral 3135   class class class wbr 5057   E cep 5457   Fr wfr 5504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-fr 5507
This theorem is referenced by: (None)
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