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Theorem dford2 7321
Description: Assuming ax-reg 7306, 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 4395 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 7316 . . . . 5  |-  _E  Fr  A
3 dfwe2 4573 . . . . 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 884 . . . 4  |-  (  _E  We  A  <->  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) )
5 epel 4308 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
6 biid 227 . . . . . 6  |-  ( x  =  y  <->  x  =  y )
7 epel 4308 . . . . . 6  |-  ( y  _E  x  <->  y  e.  x )
85, 6, 73orbi123i 1141 . . . . 5  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
982ralbii 2569 . . . 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 240 . . 3  |-  (  _E  We  A  <->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
1110anbi2i 675 . 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 240 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 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   Tr wtr 4113    _E cep 4303    Fr wfr 4349    We wwe 4351   Ord word 4391
This theorem is referenced by:  celsor  25111  ordelordALT  28301  ordelordALTVD  28643
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-13 1686  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  ax-un 4512  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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