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Theorem dford2 7577
Description: Assuming ax-reg 7562, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
Assertion
Ref Expression
dford2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
Distinct variable group:    x, y, A

Proof of Theorem dford2
StepHypRef Expression
1 df-ord 4586 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 7572 . . . . 5  |-  _E  Fr  A
3 dfwe2 4764 . . . . 5  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) ) )
42, 3mpbiran 886 . . . 4  |-  (  _E  We  A  <->  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) )
5 epel 4499 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
6 biid 229 . . . . . 6  |-  ( x  =  y  <->  x  =  y )
7 epel 4499 . . . . . 6  |-  ( y  _E  x  <->  y  e.  x )
85, 6, 73orbi123i 1144 . . . . 5  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
982ralbii 2733 . . . 4  |-  ( A. x  e.  A  A. y  e.  A  (
x  _E  y  \/  x  =  y  \/  y  _E  x )  <->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
104, 9bitri 242 . . 3  |-  (  _E  We  A  <->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
1110anbi2i 677 . 2  |-  ( ( Tr  A  /\  _E  We  A )  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) ) )
121, 11bitri 242 1  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    \/ w3o 936   A.wral 2707   class class class wbr 4214   Tr wtr 4304    _E cep 4494    Fr wfr 4540    We wwe 4542   Ord word 4582
This theorem is referenced by:  ordelordALT  28624  ordelordALTVD  28981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586
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