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Theorem efrunt 25154
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  ->  A. x  e.  A  -.  x  e.  x )
Distinct variable group:    x, A

Proof of Theorem efrunt
StepHypRef Expression
1 frirr 4551 . . 3  |-  ( (  _E  Fr  A  /\  x  e.  A )  ->  -.  x  _E  x
)
2 epel 4489 . . 3  |-  ( x  _E  x  <->  x  e.  x )
31, 2sylnib 296 . 2  |-  ( (  _E  Fr  A  /\  x  e.  A )  ->  -.  x  e.  x
)
43ralrimiva 2781 1  |-  (  _E  Fr  A  ->  A. x  e.  A  -.  x  e.  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   class class class wbr 4204    _E cep 4484    Fr wfr 4530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-eprel 4486  df-fr 4533
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