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Theorem wecmpep 4815
Description: The elements of an epsilon well-ordering are comparable. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
wecmpep  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )

Proof of Theorem wecmpep
StepHypRef Expression
1 weso 4814 . 2  |-  (  _E  We  A  ->  _E  Or  A )
2 solin 4767 . . 3  |-  ( (  _E  Or  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  _E  y  \/  x  =  y  \/  y  _E  x ) )
3 epel 4737 . . . 4  |-  ( x  _E  y  <->  x  e.  y )
4 biid 236 . . . 4  |-  ( x  =  y  <->  x  =  y )
5 epel 4737 . . . 4  |-  ( y  _E  x  <->  y  e.  x )
63, 4, 53orbi123i 1187 . . 3  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
72, 6sylib 196 . 2  |-  ( (  _E  Or  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
81, 7sylan 469 1  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    \/ w3o 973    e. wcel 1842   class class class wbr 4395    _E cep 4732    Or wor 4743    We wwe 4781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-eprel 4734  df-so 4745  df-we 4784
This theorem is referenced by:  tz7.7  5436
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