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Theorem wecmpep 4823
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 4822 . 2  |-  (  _E  We  A  ->  _E  Or  A )
2 solin 4775 . . 3  |-  ( (  _E  Or  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  _E  y  \/  x  =  y  \/  y  _E  x ) )
3 epel 4746 . . . 4  |-  ( x  _E  y  <->  x  e.  y )
4 biid 236 . . . 4  |-  ( x  =  y  <->  x  =  y )
5 epel 4746 . . . 4  |-  ( y  _E  x  <->  y  e.  x )
63, 4, 53orbi123i 1178 . . 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 471 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 369    \/ w3o 964    e. wcel 1758   class class class wbr 4403    _E cep 4741    Or wor 4751    We wwe 4789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-eprel 4743  df-so 4753  df-we 4792
This theorem is referenced by:  tz7.7  4856
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