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Theorem ordeq 4355
Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
ordeq  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )

Proof of Theorem ordeq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 treq 4091 . . 3  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
2 raleq 2665 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  Tr  x  <->  A. x  e.  B  Tr  x ) )
31, 2anbi12d 470 . 2  |-  ( A  =  B  ->  (
( Tr  A  /\  A. x  e.  A  Tr  x )  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) ) )
4 dford3 4350 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
5 dford3 4350 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
63, 4, 53bitr4g 222 1  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   A.wral 2448   Tr wtr 4085   Ord word 4345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-in 3127  df-ss 3134  df-uni 3795  df-tr 4086  df-iord 4349
This theorem is referenced by:  elong  4356  limeq  4360  ordelord  4364  ordtriexmidlem  4501  2ordpr  4506  issmo  6264  issmo2  6265  smoeq  6266  smores  6268  smores2  6270  smodm2  6271  smoiso  6278  tfrlem8  6294  tfri1dALT  6327
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