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Theorem efrunt 24074
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 4386 . . 3  |-  ( (  _E  Fr  A  /\  x  e.  A )  ->  -.  x  _E  x
)
2 epel 4324 . . 3  |-  ( x  _E  x  <->  x  e.  x )
31, 2sylnib 295 . 2  |-  ( (  _E  Fr  A  /\  x  e.  A )  ->  -.  x  e.  x
)
43ralrimiva 2639 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 1696   A.wral 2556   class class class wbr 4039    _E cep 4319    Fr wfr 4365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-fr 4368
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