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Theorem efrunt 31818
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 5195 . . 3 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥 E 𝑥)
2 epel 5136 . . 3 (𝑥 E 𝑥𝑥𝑥)
31, 2sylnib 317 . 2 (( E Fr 𝐴𝑥𝐴) → ¬ 𝑥𝑥)
43ralrimiva 3068 1 ( E Fr 𝐴 → ∀𝑥𝐴 ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2103  wral 3014   class class class wbr 4760   E cep 5132   Fr wfr 5174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-eprel 5133  df-fr 5177
This theorem is referenced by: (None)
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