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Theorem dford5reg 30001
Description: Given ax-reg 8052, 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 5413 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 8062 . . . 4  |-  _E  Fr  A
3 df-we 4784 . . . 4  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  _E  Or  A ) )
42, 3mpbiran 919 . . 3  |-  (  _E  We  A  <->  _E  Or  A )
54anbi2i 692 . 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 367   Tr wtr 4489    _E cep 4732    Or wor 4743    Fr wfr 4779    We wwe 4781   Ord word 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-reg 8052
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-eprel 4734  df-fr 4782  df-we 4784  df-ord 5413
This theorem is referenced by: (None)
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