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Theorem ordeq 4393
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 4125 . . 3  |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B )
)
2 raleq 2686 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  Tr  x  <->  A. x  e.  B  Tr  x ) )
31, 2anbi12d 473 . 2  |-  ( A  =  B  ->  (
( Tr  A  /\  A. x  e.  A  Tr  x )  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) ) )
4 dford3 4388 . 2  |-  ( Ord 
A  <->  ( Tr  A  /\  A. x  e.  A  Tr  x ) )
5 dford3 4388 . 2  |-  ( Ord 
B  <->  ( Tr  B  /\  A. x  e.  B  Tr  x ) )
63, 4, 53bitr4g 223 1  |-  ( A  =  B  ->  ( Ord  A  <->  Ord  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   A.wral 2468   Tr wtr 4119   Ord word 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-in 3150  df-ss 3157  df-uni 3828  df-tr 4120  df-iord 4387
This theorem is referenced by:  elong  4394  limeq  4398  ordelord  4402  ordtriexmidlem  4539  2ordpr  4544  issmo  6317  issmo2  6318  smoeq  6319  smores  6321  smores2  6323  smodm2  6324  smoiso  6331  tfrlem8  6347  tfri1dALT  6380
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