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Theorem dford5reg 29141
Description: Given ax-reg 8030, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
Assertion
Ref Expression
dford5reg  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  Or  A ) )

Proof of Theorem dford5reg
StepHypRef Expression
1 df-ord 4887 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 8041 . . . 4  |-  _E  Fr  A
3 df-we 4846 . . . 4  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  _E  Or  A ) )
42, 3mpbiran 916 . . 3  |-  (  _E  We  A  <->  _E  Or  A )
54anbi2i 694 . 2  |-  ( ( Tr  A  /\  _E  We  A )  <->  ( Tr  A  /\  _E  Or  A
) )
61, 5bitri 249 1  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  Or  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   Tr wtr 4546    _E cep 4795    Or wor 4805    Fr wfr 4841    We wwe 4843   Ord word 4883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-reg 8030
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-eprel 4797  df-fr 4844  df-we 4846  df-ord 4887
This theorem is referenced by: (None)
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