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

Theorem dford5reg 29410
Description: Given ax-reg 8036, 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 4890 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  _E  We  A ) )
2 zfregfr 8046 . . . 4  |-  _E  Fr  A
3 df-we 4849 . . . 4  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  _E  Or  A ) )
42, 3mpbiran 918 . . 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 4550    _E cep 4798    Or wor 4808    Fr wfr 4844    We wwe 4846   Ord word 4886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-eprel 4800  df-fr 4847  df-we 4849  df-ord 4890
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator