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Theorem dford2 7941
Description: Assuming ax-reg 7922, 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 4833 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 7933 . . . . 5  |-  _E  Fr  A
3 dfwe2 6506 . . . . 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 909 . . . 4  |-  (  _E  We  A  <->  A. x  e.  A  A. y  e.  A  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) )
5 epel 4746 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
6 biid 236 . . . . . 6  |-  ( x  =  y  <->  x  =  y )
7 epel 4746 . . . . . 6  |-  ( y  _E  x  <->  y  e.  x )
85, 6, 73orbi123i 1178 . . . . 5  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
982ralbii 2840 . . . 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 249 . . 3  |-  (  _E  We  A  <->  A. x  e.  A  A. y  e.  A  ( x  e.  y  \/  x  =  y  \/  y  e.  x ) )
1110anbi2i 694 . 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 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    \/ w3o 964   A.wral 2799   class class class wbr 4403   Tr wtr 4496    _E cep 4741    Fr wfr 4787    We wwe 4789   Ord word 4829
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-8 1760  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  ax-un 6485  ax-reg 7922
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-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  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-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-eprel 4743  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833
This theorem is referenced by:  ordelordALT  31599  ordelordALTVD  31958
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