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Theorem ordeq 4135
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 3889 . . 3  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
2 raleq 2550 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  Tr  x  <->  A. x  e.  B  Tr  x ) )
31, 2anbi12d 457 . 2  |-  ( A  =  B  ->  (
( Tr  A  /\  A. x  e.  A  Tr  x )  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) ) )
4 dford3 4130 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
5 dford3 4130 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
63, 4, 53bitr4g 221 1  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   A.wral 2349   Tr wtr 3883   Ord word 4125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129
This theorem is referenced by:  elong  4136  limeq  4140  ordelord  4144  ordtriexmidlem  4271  2ordpr  4275  issmo  5937  issmo2  5938  smoeq  5939  smores  5941  smores2  5943  smodm2  5944  smoiso  5951  tfrlem8  5967  tfri1dALT  6000
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