Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dford5reg Structured version   Unicode version

Theorem dford5reg 25411
Description: Given ax-reg 7562, 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 4586 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 7572 . . . 4  |-  _E  Fr  A
3 df-we 4545 . . . 4  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  _E  Or  A ) )
42, 3mpbiran 886 . . 3  |-  (  _E  We  A  <->  _E  Or  A )
54anbi2i 677 . 2  |-  ( ( Tr  A  /\  _E  We  A )  <->  ( Tr  A  /\  _E  Or  A
) )
61, 5bitri 242 1  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  Or  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   Tr wtr 4304    _E cep 4494    Or wor 4504    Fr wfr 4540    We wwe 4542   Ord word 4582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-eprel 4496  df-fr 4543  df-we 4545  df-ord 4586
  Copyright terms: Public domain W3C validator