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Theorem efrunt 24059
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 4370 . . 3  |-  ( (  _E  Fr  A  /\  x  e.  A )  ->  -.  x  _E  x
)
2 epel 4308 . . 3  |-  ( x  _E  x  <->  x  e.  x )
31, 2sylnib 295 . 2  |-  ( (  _E  Fr  A  /\  x  e.  A )  ->  -.  x  e.  x
)
43ralrimiva 2626 1  |-  (  _E  Fr  A  ->  A. x  e.  A  -.  x  e.  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   class class class wbr 4023    _E cep 4303    Fr wfr 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352
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