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Theorem dford2 7337
Description: Assuming ax-reg 7322, 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 4411 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 7332 . . . . 5  |-  _E  Fr  A
3 dfwe2 4589 . . . . 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 4324 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
6 biid 227 . . . . . 6  |-  ( x  =  y  <->  x  =  y )
7 epel 4324 . . . . . 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 2582 . . . 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   Tr wtr 4129    _E cep 4319    Fr wfr 4365    We wwe 4367   Ord word 4407
This theorem is referenced by:  celsor  25214  ordelordALT  28600  ordelordALTVD  28959
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-13 1698  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  ax-un 4528  ax-reg 7322
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 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-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411
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