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Theorem ordeq 4199
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 3942 . . 3  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
2 raleq 2562 . . 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 4194 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
5 dford3 4194 . 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 1289   A.wral 2359   Tr wtr 3936   Ord word 4189
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193
This theorem is referenced by:  elong  4200  limeq  4204  ordelord  4208  ordtriexmidlem  4336  2ordpr  4340  issmo  6053  issmo2  6054  smoeq  6055  smores  6057  smores2  6059  smodm2  6060  smoiso  6067  tfrlem8  6083  tfri1dALT  6116
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